2-2
Content
SOME DETERMINANTS INVOLVING POWERS OF FIBONACCI NUMBERS
BROTHER U. A L F R E D
St. Mary's College, California
In the October* 1963 issue of this journal [l] , the author discussed some of the periodic properties of Fibonacci summations.
It
was noted that a certain determinant was basic to these considerations.
Its main characteristics and value were indicated and a pro-
mise was given of additional explanation in some later issue of the
Fibonacci Quarterly.
The purpose of this article is to set forth the
manner of evaluating these determinants on an empirical basis.
The
proof of the general validity of the results obtained is to be found in
an article by Terry Brennan in this issue of the Quarterly [2],
To fix ideas the determinant of the sixth order will be used.
Written in a form that brings out its
Fibonacci characteristics it
would be:
3 3
11
1214
l l
2214
2 I
4 2
3 2"
3 3
21
3 3
3 2
3224
325
5
5 3
5432
3 3
5 3
5Z34
53
5 6 - :L
855
8452
3 3
3 5
8254
8 55
136+86-]L
5
13 8
13482
3 3
13 8
13284
16
Z
6
3
6
5
6
86
; iV
+
1 6 - ]L
+
l
6
-] L
251
2412
+
2
6
- ]L
352
+
3
6
-]
I
+
151
5
13 8
5
5
5
A c e r t a i n subtlety should be noted i n the f i r s t line a s the f i r s t " 1 "
stands for F - and the second f o r F , .
By s e p a r a t i n g t h e t e r m s
of t h e f i r s t
column into groups, t h e
p r o b l e m c a n b e c h a n g e d t o t h a t of e v a l u a t i n g t h r e e d e t e r m i n a n t s w i t h
f i r s t c o l u m n s a s i n d i c a t e d be low:
81
82
SOME DETERMINANTS INVOLVING POWERS
(1)
(2)
1
1
26
1
April
(3)
13
D e t e r m i n a n t s (1) and (2) can be evaluated in t e r m s of what shall be
called the BASIC POWER DETERMINANT.
D e t e r m i n a n t (3) will be
developed in t e r m s of the c o f a c t o r s of the f i r s t column which involve
the basic power d e t e r m i n a n t m i n u s one of its r o w s .
BASIC POWER DETERMINANT
The f i r s t d e t e r m i n a n t h a s a common factor in each of its r o w s
(the f a c t o r s a r e 1 , 2 , 3 , 5 , 8 , 1 3 r e s p e c t i v e l y ) . If t h e s e f a c t o r s be taken
out of the d e t e r m i n a n t ,
determinant.
1
1
1
1
5
24
23
22
2
5
342
4
5 3
4
8 5
4
13 8
3322
3223
3
5332
5233
5 • 3"
8352
8253
8
13 3 8 2
13 2 8 3
1
13 • 8'
5
5
13
we have what will be called the b a s i c power
F o r the sixth o r d e r , it is a s shown below:
5
This d e t e r m i n a n t is a s p e c i a l c a s e of the m o r e g e n e r a l d e t e r m i n a n t
in which the f i r s t row s t a r t s with any F i b o n a c c i n u m b e r w h a t s o e v e r
1964
OF FIBONACCI NUMBERS
4
F F. ,
F5
F?F?,
i l-l
1
5
F
* i+1
5
F
F3+1F2
F
i+1 i
1+1 1
4
F F
i+5 i+4
F
i+5
3 2
F F
i+5 i+4
4
F.
1 1-1
i l-l
4
83
i» 1
i
«*t
F2 F 3
i+1 i
F.
2
3
F F
i+5 i+4
4
F F
i+5 i+4
F5
1-
1
F5
1
5
F
i+4
The basis for evaluating this determinant is the r e l a t i o n
F F
- F
F
= (-l)n+1F
n n+k+1
n+1 n+k K }
*k
To evaluate the determinant we proceed to p r o d u c e z e r o s in the f i r s t
row.
This i s done by multiplying the f i r s t column by F . , and sub-
tracting from this F . times the second column; then multiplying the
second column by F . , and subtracting from this
column; etc.
F . t i m e s the t h i r d
The operations for the f i r s t and second columns would
be as follows:
4
F
i
r+i<Fi-iFi+i
- w
f+2<Fi-lFi+2
-
F
.4
F
and so on.
/T_
^
i+3<Fi-lFi+3'.-
F F
'"
\Wijr2Ft+2..
i i+l)
^w
F F
4
= t-1) F i F i+i •...
i i+2)
=
;
^
lA
i„ ^4
"
i+3
It is clear that the second row would have a factor
the third a factor
F~, etc.
F, ,
Thus, after eliminating the common fac-
tors and expanding by the non-zero t e r m in the last column of the
row, the absolute value of the resulting d e t e r m i n a n t would, be F , F?
•F^F.Fp. multiplied by the basic power d e t e r m i n a n t of the fifth o r d e r .
If we adopt the notation
An
84
SOME DETERMINANTS INVOLVING POWERS
ApriL
to r e p r e s e n t the b a s i c power d e t e r m i n a n t of the nth o r d e r , this r e s u l t
m a y be e x p r e s s e d a s
l*61 = n
•
(F4)
|A |
.
1=1
In g e n e r a l , for a d e t e r m i n a n t of o r d e r
l^nlvn
1
n,
(F.)-
|Aa
| .
1=1
Since the p r o c e s s m a y be r e p e a t e d , it is not difficult to a r r i v e at the
final r e s u l t :
A
=
n
Fn-X .
In the p a r t i c u l a r c a s e of o r d e r six,
It is i n t e r e s t i n g to note that the values of t h e s e b a s i c power d e t e r m i n a n t s a r e independent of w h e r e we s t a r t in the F i b o n a c c i s e q u e n c e .
SIGN OF THE BASIC POWER DETERMINANT
It is i m p o r t a n t to be able to d e t e r m i n e the sign of the b a s i c d e t e r m i n a n t value i n a s m u c h a s we shall combine the values of d e t e r m i n a n t s ( l ) a n d (2) with the values of the c o f a c t o r s of d e t e r m i n a n t (3).
The c o n s i d e r a t i o n s involved a r e a bit t e d i o u s .
c a s e s a c c o r d i n g a s n i s of the form
We d i s t i n g u i s h four
4k, 4k+l, 4k+2, or 4k+3.
The
following t h r e e f a c t o r s d e t e r m i n e the outcome:
(i)
The sign introduced by expanding from the l a s t e l e m e n t in the
f i r s t row;
(ii)
The signs of the t e r m s , of the d e t e r m i n a n t r e s u l t i n g after e a c h
of the s t e p s indicated above.
plus or a l l m i n u s .
These t e r m s will be e i t h e r a l l
1964
(iii)
O F FIBONACCI NUMBERS
The sign of
A
85
which is the s e c o n d - o r d e r d e t e r m i n a n t of the
f i r s t p o w e r s of the Fibonacci n u m b e r s in the Last two r o w s .
Thus, for the sixth o r d e r d e t e r m i n a n t we have been c o n s i d e r ing,
&z
is
The final outcome is a s follows:
(i)
(ii)
F o r o r d e r 4k or 4k+l, the sign is always plus;
F o r o r d e r 4k+2 or 4k+3, the sign a g r e e s with that of A
As noted p r e v i o u s l y ,
the b a s i c power d e t e r m i n a n t enables us
to evaluate d e t e r m i n a n t s (1) and (2).
The l a t t e r can be brought to
this form by shifting the f i r s t column so that it b e c o m e s the l a s t
column.
BASIC POWER DETERMINANTS WITH ONE ROW MISSING
To evaluate the t h i r d d e t e r m i n a n t we find the cofactors of the
e l e m e n t s in the f i r s t column.
F o r the e l e m e n t in the f i r s t row, this
cofactor is a b a s i c power d e t e r m i n a n t after r e m o v i n g c o m m o n fact o r s , but for a l l the o t h e r s it is e s s e n t i a l l y a b a s i c power d e t e r m i n a n t
with one row m i s s i n g .
order
The absolute value of such a d e t e r m i n a n t of
n with a m i s sing row between the k and (k+l)st
row will be
r e p r e s e n t e d by
A
(k I k + 1)
n
'
the i m p l i c a t i o n being that the absolute value d o e s , not depend on the
p a r t i c u l a r F i b o n a c c i n u m b e r with which it s t a r t s .
When developing
such a d e t e r m i n a n t the p r o c e d u r e is the s a m e a s for the development
of the b a s i c power d e t e r m i n a n t , only in this c a s e t h e r e is a gap.
calculation for a d e t e r m i n a n t of o r d e r
The
n with a row m i s s i n g between
the t h i r d and fourth rows can be s u m m a r i z e d s c h e m a t i c a l l y in the
following m a n n e r .
The column headings a r e Fibonacci n u m b e r s .
A
table e n t r y is the power to which the F i b o n a c c i n u m b e r at the head of
the column i s being r a i s e d .
plied t o g e t h e r .
The quantities in any one row a r e m u l t i -
In the f i r s t row we have the r e s u l t of the f i r s t step in
86
SOME DETERMINANTS INVOLVING POWERS
April
the evaluation in which the order is changed from 9 to 3; in the s e c ond, the f a c t o r s r e s u l t i n g in reducing the determinant from order 8
to o r d e r 7; e t c ,
1
1
0
1
1
1
1
1
0
1
0
1
1
1
1
0
1
1
1
1
1
0
1 .
0
1
1
1
1
0
1
1
1
1
1
1
0
1
1
0
The sum of the quantities in any column gives the power of the F i b onacci n u m b e r in the determinant.
In the above case
A9(3 | 4) = FJF^FJFJFJFJFJF, .
The s a m e r e s u l t would have been obtained if the gap had been after
the sixth row.
In g e n e r a l ,
if the determinant is of order n, a gap
after the kth . row or the (n~k)th row gives the same result.
The p a t t e r n o b s e r v e d is as follows: (1) A reduction of 2 in the
p o w e r s of F-. to F
inclusive (if k is l e s s than n-k); (2) A r e duction of 1 from F, M to F , inclusive; (3) No reduction therex
k+1
n-k
'
after. If n-k is l e s s than k, the roles of k and n-k are r e v e r s e d .
Finally,
if n - k equals k (even n), there would be a reduction of 2
from 1 to k and no r e d u c t i o n thereafter.
These r e s u l t s m a y be summarized in the following formulas.
FORMULA FOR k LESS THAN n-k
A n (k ' k+1)
k
i=l
FORMULA FOR n - k
A n*(k
k+1]
. , n-k
n F^1"1 n
1
i=k+l
Fn'\
1
LESS THAN k
k
n x l
n
i=l
F.
X
n
i=n-k+l
.
n
.,,
F n " 1+1
n
i=n-k+l
n
i=k+l
x
Ff-i+1
1964
O F FIBONACCI NUMBERS
87
FORMULA FOR k EQUAL TO n - k
n/2
A (n/2 | n/2+1)
n
,n-i-l
F
i=l
n
XI
. n,,
Fn-i+l
i
T h e s e f o r m u l a s a r e not difficult of application.
However,
for the
sake of convenience (in view of future c o n s i d e r a t i o n s ) and a s a p o s sible guide to r e a d e r s the r e s u l t s for o r d e r s 12 and 13 a r e set down
in d e t a i l .
Since, h o w e v e r ,
t h e r e is s y m m e t r y in k and n - k
only
the f i r s t half need be given in each c a s e .
TABLE OF AT 0 (v k
12
£
k+1)
F
8
x
1
2
1
10
10
4
3
1
2
10
9
4
3
1
3
1
10
11
3
10
9
4
4
10
9
4
4
1
5
10
9
5
4
1
6
10
9
5
4
1
12
TABLE OF A 1 3 ( k | k+1)
k
1
F
F,
Fn
11
10
2
10
10
3
10
4
1.0
5
10
6
10
9
9
9
9
11
10
12
x
7
6
5
4
3
2
1
1
8
7
6
5
4
3
2 •
2
1
8
7
6
5
4
3
3
2
1
8
8
7
6
5
4
4
3
2
1
8
7
7
6
5
5
4
3
2
1
8
7
6
6
6
5
4
3
2
1
9
9
9
8
13
SIGN OF POWER DETERMINANT WITH ONE LINE MISSING
The c o n s i d e r a t i o n s leading to the d e t e r m i n a t i o n of the sign of
power d e t e r m i n a n t s with a line m i s s i n g a r e involved.
i s p r e c i s e l y the s a m e a s for the power d e t e r m i n a n t .
The a p p r o a c h
The r e s u l t s for
88
SOME DETERMINANTS INVOLVING POWERS
a l l values of n,
April
i ( s u b s c r i p t of the leading F i b o n a c c i n u m b e r in the
d e t e r m i n a n t ) and k (as defined for the b r e a k point but taken modulo
4) a r e listed in the following t a b l e .
k
4r
4r+l
1
-
+
2
-
-
3
+
4
+
4r+2
i odd
4r+2
i even
4r+3
i odd
+
+
-
+
-
-
+
+
'+
+
-
+
~,
4r+3
i even
-
+
EVALUATION O F THE ORIGINAL DETERMINANT
We noted p r e v i o u s l y that the o r i g i n a l d e t e r m i n a n t could be
r e p r e s e n t e d a s the sum of t h r e e s e p a r a t e d e t e r m i n a n t s (1), (2), and
(3). D e t e r m i n a n t (1) i s s i m p l y the p r o d u c t of F i b o n a c c i n u m b e r s (one
from e a c h row) by the b a s i c power d e t e r m i n a n t .
Thus for the sixth
o r d e r , the situation would be a s follows:
F
BPD
F
l
5
4
5
F's
(1)
F
2
F
3
4
F
F
1
1
1
1
3
2
5
1
1
1
5
4
1
3
1
2
F
6
7
The sign would be n e g a t i v e .
D e t e r m i n a n t (2) can be r e l a t e d to the b a s i c power d e t e r m i n a n t
by moving the f i r s t c o l u m n i n t o the l a s t position.
F o r the sixth o r d e r ,
this involves a change of sign„ Again f a c t o r s can be taken out leaving
a b a s i c power d e t e r m i n a n t .
The p a t t e r n i s a s follows:
F,
F
BPD
r-3
F
5
4
3
F's
1
1
(2)
6
5
The sign would be positive*
2
F
F
2
5
1
1
1
1
1
4
3
2
1
4
6
1964
O F FIBONACCI NUMBERS
89
To evaluate (3) we expand by the f i r s t column.
We shall d e s i g -
nate s u c c e s s i v e e l e m e n t s of the expansion, due account being taken of
a l l signs i n c l u d i n g t h e negative quantities in the f i r s t column, by s u c cessive
capital l e t t e r s : A, B, C, D, . . . . A gives r i s e to a simple
b a s i c power d e t e r m i n a n t ; B to one with a line m i s s i n g
the second line m i s s i n g
(k = 1); C with
(k = 2); e t c . However, t h e r e a r e f a c t o r s that
have to be multiplied in each c a s e . It should be noted too that we a r e
r e f e r r i n g to d e t e r m i n a n t s of the fifth o r d e r and not of the sixth.
F
l
4
3
l
3
l
F
3
2
3
2
i
3
E
2
3
F
D
F
F
F
3
2
3
3
F
2
3
4
F
F
2
2
4
F
F
5
6
F
2
2
2
1
4
3
2
2
1
F
2
3
F
4
F
F
5
6
F
1
1
1
2
2
2
1
3
3
3
2
1
F
7
1
2
2
2
F
3
F
4
F
2
1
1
1
2
3
3
3
F
3
F
4
F
F
5
F
2
1
2
1
F
5
6
6
(negative)
7
(positive)
7
F„
2
2
1
2
1
1
2
1
3
2
4
3
2
2
1
F
3
F
3
F
2
F
1
F
1
F,
F
2
2
4
3
3
3
1
2
1
1
(positive)
(negative)
1
1
(negative)
90
SOME DETERMINANTS INVOLVING POWERS
F
F
l
2
F
•
F
3
F
4
F
5
4
3
2
1
2
2
2
1
4
3
4
3
2
1
F
6
April
7
(positive
S u m m a r i z i n g in one table (omitting the f i r s t and second Fibonacci
number f a c t o r s a s they a r e both unity) we have the following for the
evaluation of the d e t e r m i n a n t of the sixth o r d e r .
Sign
F
(1)
(2)
-
4
+
A
-
B
F
3
F
4
F
5
6
1
F
7
3
2
4
3
2
1
4
3
2
2
1
3
3
3
2
i
C
+
+
3
3
3
2
1
D
-
4
3'
2
2
1
E
-
4
3
2
1
1
F
+
4
3
2
I
The following p a i r s of t e r m s combine:
1
E and (1); F and (2); A and
o
o
D; B and C. The r e s u l t i n g s u m s have a c o m m o n factor of 2 3
2
5 , the adjoint factor being 144. Thus finally the value of the sixth
12 5 2
o r d e r d e t e r m i n a n t is found to be 2
3 5 .
DETERMINANT OF ORDER 12
Without justifying a l l the i n t e r m e d i a t e s t e p s , the s u m m a t i o n
table for o r d e r 12 is shown below*
Sign
F
3
F
1
4
D
F
6
F
7
F
8
F
9
F
10
F
(1)
+
10
9
3
7
6
5
4
3
2
n
£F
F
^13
1
1
12
(2)
.
10
7
4
3
2
1
10
8
7
6
6
5
+
9
9
8
A
5
4
3
2
2
1
B
-
9
9
8
7
6
5
4
3
2
I
C
-
8
8
7
6
5
4
3
2
1
D
•t
9
9
3
4
8
7
7
6
5
5
4
3
2
1
1964
O F FIBONACCI NUMBERS
Sign
F
3
E
+
F
-
G
-
H
I
+
+
9
9
9
9
9
J
-
10
K
-
10
L
.+
10
F
F
4
F?
F
8
7
6
6
S
7
6
8
7
7
8
8
7
9
9
9
9
8
7
8
7
8
7
8
7
5
•
6
6
6
6
6
6
6
6
91
6
b
4
F
12
- 11
2
3
6
5
i
3
5
5
4
3
5
4
5
4
4
5
5
5
F
8
F
9
F
•Mo
13
1
2
2
1
2
1
3
3
3
2
4
3
2
4
"2,
7
2
1
1
1
4
3
2
1
It will be noted that the following p a i r s add up to z e r o :
and G; C and H; B and I; A and J; (1) and K; (2) and L.
value of the determinant is z e r o .
F
1
1
E and F; D
T h e r e f o r e , the
The s a m e r e s u l t w a s found for o r -
d e r s 4, 8, and 16.
DETERMINANT OF ORDER 1!
Sign
(1)
(2).
+
+
F
3
11
11
A
-
11
B
-
10
C
10
D
+
+
10
E
-
10
F
-
10
G
+
+
10
H
I
-
10
J
-
10
K
+
11
'•+
11
L
M
-
10
11
F
4
F
5
10 9
10 9
F
6 F?
F
8
F
9
F
F
10
ll F
8
7
6
5
'4
3
2
8
7
6
5
4
3
2
5
4
3
F
12
13
1
1
F
l
1
•'
2
1
2
1
2
I
10 9
10 9
8
7
6
8
7
6
5
4
3
'2'"
3
9
9
9
9
9
9
9
9
8
7
6
6
4
4
3
8
8
7
6
5
5
4
3' ' \2
1 '
8
7
7
6
4
3
2
1
8
7
6
7
5
4
1
7
7
6
5
2
8
8
7
6' 5
3
2
1
1
9
10 9
10 9
8
7
6
5
5
4
4
4
3
3
2
8
6
6
6
4
3
2
l'
8
7
6
5
4
3
8
7
6
5
4
3
3
2
2
i
2 . 1
10 9
10 9
8
8
7
7
6
6
5
5
4
4
3
3
,2
2
•
•
1
1
1
92
SOME DETERMINANTS INVOLVING POWERS
April
It will be noted that the following p a i r s add up to z e r o : E and G;
C and I; A and K; (2) and M.
The o t h e r s can be combined to give the
following t a b l e .
Sign
F
F
F
9
9
7
6
5
4
3
2
7
6
5
4
3
3
2
1
8
7
6
5
5
4
3
2
1
8
6
7
6
5
4
3
2
1
(1), L
B, J
+
3
12
-
11
10
D, H
+
11
F
-
10
9
9
4
10
10
11
12
1
13
F
14
1
After taking out the c o m m o n factor
10 9 F 8 7 6 6 5 4
*3
4 5 6 7 8 9 10 11 12 13 14
the following r e m a i n s for evaluation:
Sign F 3
F4
F5
F6
F?
+
2
1
1
1
1
-
1
1
1
1
1
+
1
1
1
FQ
1
F()
1
F^
F
n
F
12
F
1
1
1
1
1
1
1
1
1
1
13
F
14
No e a s y method was found for evaluating the sum of t h e s e q u a n t i t i e s .
E s s e n t i a l l y it w a s a m a t t e r of evaluating t h e m ,
then factoring.
combining t h e m and
F o r t u n a t e l y , a s the n u m b e r s to be f a c t o r e d i n c r e a s e d
in size going up to 23 digits in one i n s t a n c e , a p a t t e r n involving F i b onacci and Lucas n u m b e r s w a s d i s c o v e r e d with the r e s u l t that the
f o r m u l a s (1) and (2) on page 38 of [l] w e r e d i s c o v e r e d .
The m a t t e r can be allowed to r e s t h e r e .
The path p u r s u e d h a s
been i l l u s t r a t e d in sufficient detail to allow o t h e r s to explore t h e s e
interesting determinants.
The f o r m u l a s obtained a s well a s the d e t e r -
minant values to the twentieth o r d e r a r e set forth in the p a p e r [l] and
need not be r e p e a t e d .
REFERENCES
1.
B r o t h e r U. Alfred, " P e r i o d i c P r o p e r t i e s of F i b o n a c c i S u m m a 2.
t i o n s , The F i b o n a c c i Q u a r t e r l y , 1(1963), No. 3, pp. 3 3 - 4 2 .
T. L„ B r e n n a n " F i b o n a c c i P o w e r s and P a s c a l ' s T r i a n g l e in a
M a t r i x " The Fibonacci Q u a r t e r l y , 2(1964), No. 2, pp. 9 3 - 1 0 3 .
xxxxxxxxxxxxxxxxxxxx
FIBONACCI POWERS AND PASCAL'S TRIANGLE IN A MATRIX - PART I *
T E R R E N C E A. BRENNAN
Lockheed M i s s i l e s and Space Co., Sunnyvale, California
1.
INTRODUCTION
The m a i n p o i n t of this p a p e r is to display some i n t e r e s t i n g p r o p e r t i e s o f t h e (n+1) X (n+1) m a t r i x P
defined by imbedding P a s c a l ' s
t r i a n g l e in a s q u a r e m a t r i x :
0 0 0 1
0 0
11
0 12
P
(1.1)
The m a t r i x P
n
=
13
1
3 1
was o r i g i n a l l y c o n s t r u c t e d by the author in o r d e r
to evaluate a d e t e r m i n a n t p r e s e n t e d by B r o t h e r U. Alfred.
The d e -
t e r m i n a n t , and its origin, h a s subsequently been published in [l] and,
for the sake of c o m p l e t e n e s s ,
2.
its evaluation will be p r e s e n t e d h e r e .
THE P R O B L E M AND ITS SOLUTION
THE PROBLEM:
Evaluate the fifth o r d e r d e t e r m i n a n t
5
+ 1 - ]1
141
I3 I2
1213
1
5
5
L
2° + 1 - 1
5
5
3 + 2 - ]I
241
2312
2213
2
342
3322
3223
3
- .L
543
5332
5233
5
5
5
8 D + 5 - .L
845
8352
8253
8
1
(2.1)
5
5
5
+ 3
5
" P r e s e n t e d originally at the R e s e a r c h Conference of the Fibonacci
A s s o c i a t i o n , D e c e m b e r 15, 1962.
93
94
and its n - t h o r d e r generalization.
t h e f i r s t c o l u m n w o u l d be
u
April
FIBONACCI POWERS and PASCAL'S
,,
n+1
and
u
n
n
F o r the n~th o r d e r the p o w e r s in
and the
in the l a s t row, w h e r e
d e t e r m i n a n t would extend
u
n
to
is the n - t h F i b o n a c c i n u m -
ber:
u
n+1
(2.2)
+ u
n
with
,
n- i
u;
0
0, u ,
T h e - d e t e r m i n a n t (2. 1), w h i c h w e w i l l c a l l . D_
(D
in generaL),
w i l l be e v a l u a t e d a s a n e x p a n s i o n of c o f a c t o r s of t h e f i r s t c o l u m n . In
o r d e r to k e e p t r a c k of t e r m s i n t h e e x p a n s i o n it i s c o n v e n i e n t t o d e fine D
f o r a n a r b i t r a r y s e q u e n c e a„« a. , ' a~, . . . . a ,'
by a p J
r
n
0
1 2
n+1
p r o p r i a t e l y p l a c i n g t h e m e m b e r s of t h i s s e q u e n c e i n t h e f i r s t c o l u m n
of
D :
D 5 {a}
(2.3)
a
2
a
3
a
4 '
1i
For
a,
i
5
1
i4i
ar
,5
1 5
u a } - 1 aQ
ar ~
C l e a r l y (2. 1) i s
5
5
a
,J
a,
D r {a} w i t h
l
241
L
a^
4
3 2
3
a
4
5 3
5
a,r
845
a n = a, = a~ = . . . = a , = - 1.
s i m p l i c i t y w e w i l l c o n t e n t o u r s e l v e s w i t h t h e r e d u c t i o n of
t h e fifth o r d e r d e t e r m i n a n t
(2. 3) w h i l e m e n t i o n i n g t h e c o r r e s p o n d i n g
r e s u l t s for the g e n e r a l c a s e .
T h e r e d u c t i o n r e s t s on t h e g r o u n d w o r k
of B r o t h e r U* Alfred,,
THE SOLUTION:
.
B a s i c to t h e r e d a c t i o n i s t h e d e t e r m i n a n t of t h e f o l l o w i n g m a t r i x :
r 4
A
2 ~~
(2.4)
Br
\ .J
\ .4
j 8~
i 3 .i
I2!2
1
3
2 1
2212
2
3
? ?
3 2
3
5232
5
3 2
3
5 3
3
8 5
2
8 5
2
8
3
il
1964
TRIANGLE IN A MATRIX
95
and in g e n e r a l B . w h e r e n is the o r d e r of the m a t r i x and i de~
°
ni
notes the f i r s t row e n t r i e s a s u. ,, and u..
i+l
i
An i n t e r e s t i n g p r o p e r t y of the d e t e r m i n a n t of B - . is that its
D j J.
magnitude is independent of the index, or s t a r t i n g point,
i.
This fact
is evident when we m u l t i p l y the two m a t r i c e s
B c . QA - B c . . ,
5S I 4
5$ l+l
(2.5)
where
(2.6)
Q
4
=
1
1
1
1
1
4
3
2
1
0
6
3
1
0
0
4
1
0
0
0
1
0
0
0
0
is the m a t r i x of (1.1) " t r a n s p o s e d " about its counter diagonal.
Since
the d e t e r m i n a n t of Q . is ±1 we have
1
5, i '
5, i+l
M o r e p r e c i s e l y we can develop
l,
(2.7)
IQ,
Q
'n- 1
-i,
lQ 2 h
1,
(-i) n ( n - l ) / 2
We can s t a r t , then, with B r „ and shift indices on each row to obtain
b, (J
(2.8)
But
B
5,0Q4=B5,i
•
96
April
FIBONACCI POWERS and PASCAL'S
,4
B5,0
0
0
0
i3i
I
!
2
231
221
2
^
2
3
5^ 3
2
1 • 1*
2
? 2
5- 1
P a s s i n g to d e t e r m i n a n t s w e h a v e
* 5 f i « -= - Q4, '
where
1
B^ ~
D,
.B
- 5c i 0 l
=
t-l)
1 0 i
l-
2 - 3 - 5
h a s b e e n e x p a n d e d byJ c o f a c t o r s
0 '
common row factors have been removed.
c u r s i v e p r o c e s s for e v a l u a t i n g
|B . |
|B4>i
of i t s f i r s t r o w a n d
Having established a r e -
the g e n e r a l f o r m u l a
maybe
shown:
1B
. = (-1)
v
m'
where
u
, u
u
„ .
n - 1 n - 20
n-3
n-2
s = n(n-l)(3i + n-2)/6 .
In a n o t a t i o n w h i c h w i l l b e m o r e c o n v e n i e n t t o u s e , w e d e f i n e
S
o
= 1,
S ,. = N( - l ) n S
n+1
'
n
F~(x)
=1,
0V '
F
W (x) = F
(F(x))
for
. . (x) = x . . F v(x)
n+1
n+1 n
n > 0
f o r a n yJ s e q u e n c e / x )
^
^ n>
Then
(2.9)
Q .-l
= S
and
n
IB .1 = S 1 - 1 F (S)W , (u)
' ni'
n
n
n-1
Let us s e e w h a t p r o g r e s s c a n be m a d e with
|Dn |
.
W r i t i n g ( 2 . 3) a s
t h r e e s e p a r a t e d e t e r m i n a n t s on t h e f i r s t c o l u m n w e h a v e ,
symbolically,
1964
(2.10)
TRIANGLE IN A MATRIX
97
D 5 {a)
The f i r s t d e t e r m i n a n t is the h a r d one. The second c o m p a r e s nicely
with |Bp. , I after a common factor is r e m o v e d from each row. The
i
third becomes
[Br -. | when the f i r s t column is moved to the last (a
change in sign for an even o r d e r d e t e r m i n a n t ) and c o m m o n f a c t o r s a r e
r e m o v e d from each row.
D.H
Hence
"al
F
6< u > l B 5 , l l - a 0 F 5 < u >
lB5,l
and using (2. 9)
(2.11)
D
.{a) =
-a x F 5 (S) F 6 (u) W 4 (u)
-a Q F 5 (u) W 5 (u)
Expansion of the f i r s t d e t e r m i n a n t by cofactors of its f i r s t column
gives r i s e to d e t e r m i n a n t s of the form
April
FIBONACCI POWERS and PASCAL'S
I3
3
2
( 1 - 2 - 5 - 8 ) ' (I 8 1- 3-5)a.
(2. 12)
5
I2 1
?
Z 1
3
5
^3
the m i n o r being s i m i l a r to
ing.
2
3
82 5
1 • I2
2
2 • \L
I3
3
1*
32
33
8 • 52
53
5
| B . •, j except that the t h i r d row is m i s s -
Our t a s k is to evaluate these m i n o r s .
Starting with
BJ 5. , I
(2.4)we form Ihe p r o d u c t
3
U
n3
0
B
2. 13)
5, 1
Q
4
1
u ,
3
0
-2
2
u ,
2
2
131
U
n2
I
0
~l
0
2
2 1
-l
U
-2
0 • u"
1 • 0"
4~1
u_2l
-1
4
2
1 1
22 I 2
3
U
2 • r
1"
H e r e , as in (2.8), the m a t r i x Q shifts indices on each r o w a n d , o v e r applying this shift,
i n t r o d u c e s the negative side of the Fibonacci s e -
quence by way of the r e l a t i o n u
, = u ,, - u . Expanding the d e t e r m i -
nant of (2. 13) by the t h i r d row we have
(2.14)
|B5>1i
|Q4I"3= ( - u V ^ u ^ H i ^
u2)|B4(_2|
where
u
B~ _ is B „ „ v( i . e . , the fourth o r d e r m a t r i x of (2.4) with
4,-2
4, -2
and u 7 in its f i r s t row) and w h e r e the s u p e r s c r i p t 3 denotes
that the t h i r d row is m i s s i n g .
x
(2.15)
B4,-2
3
-l
u
2
-l
3
n2
u
-2
u
-l
u
2
-2
U
3
~2
0
0 u__,
. . 2
0 u ,
3
u^ 1
3
1 .
2
1 0
1* 1
2
1-0
1 . I2
3
0J
l3
1964
TRIANGLE IN A-MATRIX
99
We t r a n s f o r m (2. 15) to the d e s i r e d m a t r i x of (.2. 12) by
< 2 - l6 >
B
P a s s i n g to d e t e r m i n a n t s ,
1
4,i'
l-2Q3 =
B
ll
and combining (2.16) with. (2.14) we have
luTj^TT^r^? |Q P ' 5, I ' *
Using° u - n = ~ ( - l ) u n for the negative
half of the Fibonacci sequence,
©
•
-i
and evaluating the known d e t e r m i n a n t s ^
W U)
, 3 .
12
^
l
;
V
I B 4! , 1, ' | = (-1)
-S- S.
"5
3 F,(S)
4 ' F2(u)F^(u)
The g e n e r a l c a s e , using this technique, rric,y be formulated a s
(2. 17)
\Bl J = ( - D n r S n + 1 S r F n (S)
W (u)
w
r-1
- ^
r
— —
n+1 -r(u)
Two simplifications to (2. 17) a r e in o r d e r :
(-1)
x
S ,, S = S ,,
n+1 r
n+l-r
and
W (u)
F (u)
vu;
F~~T(u)
F nx+l l - r xTuT "- Wn - l,{u)
F r - 1. (TITF™77™T5T
r-lx
n+l-r
It s e e m s a.rr
p p r o pr r i a t e , since F \(u)/ = u • un - I , un - 2 - . . . u2. u1, is
n
n
a f a c t o r i a l type p r o d u c t for the ssequence / u }* that we define the
s p e c i a l i z e d "binomial coefficient"
100
FIBONACCI POWERS and PASCAL'S
April
m • F M V. W = '.M-'-t:]-' •
n-4x
r
We t h e n h a v e
| B r , | = F (S) W AM) S , ,
f n1] .
1
x
v
n, 1 '
n '
n - l ' n + l - r L^-IJ
T h e r e m a i n i n g d e t e r m i n a n t of (2. 10) m a y n o w b e e x p a n d e d , a n d t h e
general case h a s the form
n+1
F J_1 (u)
F (u)
S (-l) r a r J^LL^LL
U
U
r=2
r
r-1
'
r
|B
1 J
n
l l
~ '
or
F
, ( S ) W %(u)
n-1
n '
n+1
S
r=2
v(-I)
1
'
"
S ,,
[~n M a
n+l-r L r J r
.
T h e f i r s t t w o d e t e r m i n a n t s (2. 11) r o u n d o u t t h e s u m m a t i o n n i c e l y f o r
k=l
a n d 2, s o that w e c a n s t a t e
D v{a} = F . ( S ) W x(u)
n >
n-P
n '
or,
n+1
2
A
r=0
( - l ) r S ,.
n+l-r
summing backwards^
(2.18)
Dn{a} M - i r
1
F n _ l ( S ) Wn(u)
"z
r=0
(-l)rSr f ^
At this point w e c o n s i d e r t h e s u m m a t i o n
/:>
T
1 a
L r J
r
0
1QX
(2. 19)
{a} =
n v/
n+1
2
( - l ) r S ["
l a ,.
_0
r L r J n+l-r
and the a s s o c i a t e d polynomial
(2.20)
*
n ( x )
=
n
z
r=0
(
.
1 }
r
s
[n+ljxa+l-r
1
] an+1.
r
1964
TRIANGLE IN A MATRIX
101
F o r the f i r s t few values of n we have
* 0 (x) = x - l ,
2
0 x (x) = x - x - 1 ,
* 2 (x) = x 3 - 2 x 2 - 2 x + l = ( x + l ) ( x 2 - 3 x + l ) ,
(2. 21)
4
3
2
2
?
0 3 (x) = x -3x -6x +3x+l = (x +x-l)(x - 4 x - l ) ,
« 4 (x) = x 5 - 5 x 4 - 1 5 x 3 + 1 5 x 2 + 5 x - l = (x- 1 )(x 2 +3x+l )(x 2 -7x+l)
The f a c t o r i z a t i o n s suggest the r e l a t i o n
<6n(x) = ( - l ) n _ i (x 2 - v n x + ( - l ) n ) 0 n . 2 ( - x )
(2.22)
where
v
is a Lucas n u m b e r , and v = u ,, + u , . (2. 22) m a y
n
n
n+1
n-1
be p r o v e d by induction, and the complete f a c t o r i z a t i o n of 0
comes
from the identity
v
n
= a
+b
,
where a
- a - 1= b
- b - 1 =0
Thus (2. 22) b e c o m e s
0 n (x) = (-l) 1 1 " 1 ( x - a n ) ( x - b n ) 0 n _ 2 ( " x )
=
<ab)n"1
• ( x - a n ) ( x - b n ) 0 n _ 2 (x/ab) ,
and we can c o n s t r u c t
0 (x) =
n
The evaluation of
(2. 18). and (2. 19),
D n {a}
for
n
n
(x-a b
r =0
) .
a Q = 3L^ = . . . = - 1
b e c o m e s , from
102
F I B O N A C C I P O W E R S and P A S C A L ' S
(2.24)
D (a) = (-1)
n l '
The
evaluation
F AS) W (u) 0 ( 1 )
n-1
n
n
of 0 (1)
requires
separate c a s e s ,
Using (2.23) with
(2.25)
n
?
XI ( l - ( - l ) n " r a
r=0
0(1)=
n
the
April
.
investigation
of four
b = - l/a
)= 0
if and o n l y i f n = 4k
When n = 4k + 2 t h e q u a d r a t i c f a c t o r i z a t i o n (2. 22) b e c o m e s
n/2
0n(x) = (i+x)
n
?
(x + ( - i ) r v 2 r + i)
r=l
and
0n(l) = 2
n/2
n
r=l
(v2r + 2(-l)r) .
U s i n g the w e l l k n o w n r e l a t i o n
(2. 26)
For
0 (1) = 2
n
2
v_ = v ?
n/2
7
II v "
, r
r= 1
when
r
+ 2(-l)
we have
n = 4k + 2 .
n = 4k ± 1 w e h a v e , f r o m (2, 22)
*n(x) =
2
II
r=0
( x Z - ( - l ) r + 1 v 2 r + 1 x - 1) ,
when
n = 4k - 1
( x 2 + ( - l ) r + 1 v 2 r + 1 x - 1) ,
when
n = 4k + 1
n-1
0 (x) =
n
r=0
s o that
2 27
< '
>
n--1
"~2~~
^ >
=
2k-1
n
" ^^Zr-l^Zk-l
n
r=0
r=0
when
n = 4k - 1
V
2r+1
'
1964
TRIANGLE IN A MATRIX
103
n-1
T"
,,
2k
r
0 ( i , , n (-D v 2 r + 1 = s 2 k + 1 n
(2.28)
r=0
when
Combining (2. 25),
v2r+1
r=0
(2* 26),
n = 4k 4- 1 .
(2. 27),
(2. 28) with {2. 23), and using the
sign convention
S
= (-i) 1 ^ 1 1 " 1 )/ 2
and
F (S) = -1
only when
n = 4k + 2 ,
we have
D
4k=°>
2 k
k
D
4k + 1
=
W
^
U
4k+l< >
n
V
n
2r+1 ,
r=0
2k-1
k
D
4k-1
= (
l)
~
W
4k-l
(u)
n
V
n
2r+1 ,
r=0
2k-1
D
4k + 2
= 2 W
u
4k»-2< >
n
n
r=0
V
r+1 f
„ / v
n n-1 n~2
2
W W = u u
u~
... u , u =
n
1 2
3
n-1 n
_
II
,
r=l
n+l-r
u
r
REFERENCES
1.
Brother U. Alfred, Periodic properties of Fibonacci summations, Fibonacci Quarterly, 1(1963), No, 3, pp. 33-42.
Continued next issue.
xxxxxxxxxxxxxxxxxxxx
April
104
FIBONACCI GEOMETRY
H. E. Huntley
Below a r e s o m e additional o b s e r v a t i o n s about H u n t e r ' s
article.
w+z-jL_JL_—E
Y
[1]
.- -- 5
If the r e c t a n g l e A B C D h a s a t r i a n g l e /
D P P ' i n s c r i b e d within it so that
AAPD
= V|
ABPP' = AP'DC then x(w+z) = wy = z(x+y)
v^hence
y __ w+z _ z
x
w
w-z
(i)
(ii)
2 2
.
.*.w -z = wz, l. e. ,
F r o m (i)
(iii)
2
2
w -wz-z = 0
+ V7+4z
^>z
~ = *P or y = ^?x
Thus P , P ' divide t h e i r sides in the Golden Section.
Now, suppose ABCD is the Golden Rectangle,
beloved of the
G r e e k a r c h i t e c t s , i.e., A B / B C = <P, then ^ - ^ = <p. Hence, from (ii)
,
x(l + <p)
i. e. x= <pz whence x= w. F r o m (i) y = w+z = (p z.
and(lll)
::=
TfT+?]
*
Since < A = < B r t < and x = w, y = w+z, t r i a n g l e s PAD, P ' B P a r e
congruent. It follows that P D = P P 1 , t h a t < A P D is the c o m p l e m e n t
of < B P ? ' , whence < D P P ' is a right angle.
The a r e a of the right t r i a n g l e is
J(w2+y2)
=
J ( ^ 2 z 2 + / z 2 ) = i / z 2 ( ^ 2 + l ) = iz2(^+l)(^+2)=iz2(4^+3)
we m a y conclude, t h e r e f o r e , that if the r e c t a n g l e is the Golden R e c tangle, that i s , if its adjacent sides a r e in the Golden Ratio, <p, then
the i n s c r i b e d t r i a n g l e is r i g h t - a n g l e d and i s o s c e l e s , the length of the
equal sides being z "w4^+3 .
E d i t o r i a l Note: P P 1 1(1 AC
1.
REFERENCES
Jo A. H. Hunter, " T r i a n g l e I n s c r i b e d in a R e c t a n g l e " 1(1963) Oct o b e r , pg. 66,
ON SUMMATION FORMULAS FOR FIBONACCI AND LUCAS NUMBERS
DAVID Z E I T L I N , Honeywell,
Minneapolis, Minnesota
n
X F , , , where
, ,
ak-b
k=l
a > b a r e positive i n t e g e r s and F, a r e Fibonacci n u m b e r s with
F nu = 0, F ,1 = 1, and F n+2
. 0 = Fn+1
In this note,
(1 + F , n = 0, 1, . . . .
n
we will e s t a b l i s h a m o r e g e n e r a l s u m m a t i o n formula which yields the
Recently, Siler [ l ] gave a closed form for
'
r e s u l t of [ l ] a s a s p e c i a l c a s e .
G e n e r a l s u m m a t i o n f o r m u l a s for F i b -
onacci and Lucas n u m b e r s will be obtained as s p e c i a l c a s e s of our
general result.
Theorem.
Let p, q, u * and u,
be a r b i t r a r y r e a l n u m b e r s ,
and Let
(i)
n+2
n+1
pun
(n = 0, 1, . . . ) ,
n . n
S = r + r
(n= 0 , 1 , . . . ) ,
1
2
n
2
2
w h e r e r , / r~ a r e r o o t s of x - qx + p = 0 (i. e. , q - 4p •/ 0). We define
(2)
(3)
u_n = (uQSn - u n ) / p n
(n= 1 , 2 . . . . . ) ,
(4)
S
(n= 1,2, . . . ) .
-n=Sn/pn
Let a = 0, 1, . . .• d = 0, ±1, ±2, . . . , and let x be a r e a l n u m b e r .
71 „
(1-S
x
a
, a 2.
x + rp x )
v
2
in
k=
(5)
- x
u , ,,x
ak+d
k
Then
a n+2
= rp x
u . j
an+d
°
u•an+a+d
, , -j + xu a+d
, j + x(1-xSa ) ud,
.
M o r e o v e r , in the r e g i o n of c o n v e r g e n c e , we have
a 2 ' *°
k
(1-Sax+P x ) S uak+dx =ud + (ua+d- u d S a ) x
(6)
.
k=0
Proof. If C , i = 1,2, a r e a r b i t r a r yJ c o n s t a n t s , then u = C, r 1 + C 0 r ,
—
l
n
1 1
2 Z
n = 0, 1, . . . , is the g e n e r a l solution of (1). Then
v
k = u ak+d
=
(Clrl)(rl)k
+
<C2r2)(r2)k '
s a t i s f i e s the l i n e a r difference equation
105
k = 0, 1, . . . ,
106
ON S U M M A T I O N F O R M U L A S F O R F I B O N A C C I
<7>
V
s i n c e (x
k+2 =
S
aVk+l "P^k
(*=0,1....)
- S o x + p ) = (x - ^ ( x - r ^ ) .
a
k
S
V x
. k=0
M u l t i p l y i n g b o t h s i d e s of (7) by x
k+2
and then s u m m i n g both s i d e s with
k3 w e o b t a i n
T .0
n
kfZ
_ .
v
= x &
k,Z0 Vk+2X
a = V k + l
•
.
-
r e s p e c t to
n
<»>
.
Let
n
g(x) =
April
, .,
k+1
X
0
a 2
- P
X
n
v
,*
k=0
V
,.
. k
kX
'
k=0
We n o t e t h a t
x
,~,
(9)
k+2
^
vk+2x
k= 0
(10)
=
k=0
V
k+lX
If w e s u b s t i t u t e
, , .
=gWtv
=
^
x ) + V
6 + 2
n+2 .
n+1
x.
-v
+ v n + 1 x
n+l
X
-
V
(9) a n d (10) i n t o (8), u s e (7) to e l i m i n a t e
The g e n e r a t i n g function for
R>0
for
|x|<R^ v x
Remarks.
-v
and suppose that
—>0 a s
s
k=0
v,
V
,
v
?s
(5).
c o n v e r g e s for
k
n->c&<.Thus, for
|x|<R.
Then, f o r u n = 0 a n d u , = 1, w e
~ F , F
= (-1)
F , and S = L , the well-known
n
n
-n
n
n
n
s e q u e n c e , w h e r e L n = 2 a n d L, = 1.
T h u s , (5) a n d (6), f o r u
become, respectively,
(x 1 - L x + ( - l ) a x 2 ) £ F . , , x k = ( - i ) a x n + 2 F
,,
a
i n
ak+d
an+d
k=0
-x
(12)
n+1
F
, , , + x F , , + ( 1 - x L ) F , .,
an+a+d
a+d
a d
(1-Lax + (-l)ax2)
Tiien,
| x | < R , (5) y i e l d s (6) a s n ^ o o .
have u
(11)
and
i s r e a d i l y o b t a i n e d f r o m (5).
L- X
L e t q = 1 a n d p = - 1 in (1).
Q
0 *
s o l v e f o r g(x), w e o b t a i n o u r p r i n c i p a l r e s u l t ,
Let
l X
~
Fak+dxk = Fd + (Fa+d - FdLa)x
k=0
Lucas
= F ,
1964
AND LUCAS NUMBERS
107
The m a i n r e s u l t of [1] is obtained from (11) for x = 1 and d = - b .
F o r x = - 1 , (11) yields the i n t e r e s t i n g r e s u l t
(13)
K-U
+
<-1)nFan+a4d-Fa+d
+
<La+1)Fd "
F o r d = 0, (12) yields
(l-Lax + (-l)ax2)
(14)
T Fakxk= Fax.,k=Q
Again, let q = 1 and p = ~1 in (1).
( a = 0 , 1 / . ...) .
Then, for u n = 2, and u = 1,
we now have u = L , S = L , and L = (-1) L . Thus, with u • = L ,
n
n n
n
-n
n
n
n
(5) and (6) b e c o m e , r e s p e c t i v e l y ,
(l-Lax+(-l)ax2)
(15)
(16)
x
~ '
z Lak+dxk= ("Daxn+2L
k=0
- ^n+1Lan+a+d
4- x L a+d
_ + (x l - x L a' ) dW
( 1 - L x + x( - l ); a x 2 ) IT L , , , x k = L , + x(L , , - L L , ) x .
*
a
i r* • ak+d
d
a+d
a d'
k=0
REFERENCES
1.
Ken Siler, F i b o n a c c i s u m m a t i o n s , Fibonacci Q u a r t e r l y , Vol. 1,
No. 3, October 1963, pp. .67-69..
xxxxxxxxxxxxxxxxxxxx
108
LETTER TO THE EDITOR
The Editor,
Fibonacci Quarterly.
Dear Dr0 Hoggatt,
I refer to the article, "Dying Rabbit Problem Revived" in the December
1963 issue. The solution given there is patently wrong — if only because the alleged number of rabbits tends to minus infinity as n tend
to infinity. It may easily be shown that the correct answer, X , is
given by the recurrence relation
X n o = X ., ~ T X _ . - X
n+13
n+12
ntl 1
n
3
n
0
together with the initial conditions
X = F , . for n = 0, 1, . . . , 11;
n
n+1
X, 0 = 232 .
12
2
13
12
In view of the fact that the two equations z - z - 1 = 0 and z
- z
- z•*••*• + 1 = 0 have no common root, it is clear that the answer can
never be expressed simply as a-linear expression in Fibonacci and
Lucas numbers whose coefficients are merely polynomials i n n . For,
any such expression, Y, where the highest power of n which occurs is
n m , satisfies
(E 2 - E - l ) m + 1 Y = 0 .
In particular the expression found by Bro. Alfred satisfies
(E 2 - E - I ) 2 Y = 0 .
The error made by Bro. Alfred stems from his table on p. 54 where
the number of dying rabbits in the (n+13)th month is seen to be F for
n = 1, 2, . . . 11 and it is then assumed without proof that this is true for
other values of n. In fact the very next but one value on n, namely
n = 13 shows that this is false. In fact of course the number of dying
rabbits in the (n+13)th month equals the number of bred rabbits in the
(n+l)th month, and this will be less than F for all n exceeding 12.
Yours sincerely,
(John H. E. Cohn)
BEDFORD COLLEGE
(University of London)
SQUARE FIBONACCI NUMBERS, ETC.
JOHNH.E.COHN
Bedford College, University of London, London, N . W . I .
INTRODUCTION
An old conjecture about Fibonacci n u m b e r s is that 0, 1 and 144
a r e the only p e r f e c t s q u a r e s .
Recently t h e r e a p p e a r e d a r e p o r t that
computation had r e v e a l e d that among the f i r s t million n u m b e r s in the
sequence t h e r e a r e no further s q u a r e s [ 1 ] .
This is not s u r p r i s i n g ,
a s I have m a n a g e d to p r o v e the t r u t h of the conjecture, and this s h o r t
note is w r i t t e n by invitation of the e d i t o r s to r e p o r t m y proof.
o r i g i n a l proof will a p p e a r s h o r t l y in [ Z ] and the r e a d e r is
The
referred
t h e r e for d e t a i l s . However, the proof given t h e r e is fairly long, and
although the s a m e method gives s i m i l a r r e s u l t s for the Lucas n u m b e r s , I have r e c e n t l y d i s c o v e r e d a r a t h e r n e a t e r method, which s t a r t s
with the Lucas n u m b e r s ,
p e a r s below.
and it is of this method that an account a p -
It is hoped that the full proof t o g e t h e r with its c o n s e -
q u e n c e s for Diophantine equations will a p p e a r l a t e r this y e a r .
I might
add that the s a m e method s e e m s to w o r k for m o r e g e n e r a l s e q u e n c e s
2
4
of i n t e g e r s , thus enabling equations like y = Dx + 1 to be c o m pletely solved at l east for c e r t a i n values of D.
Of c o u r s e the F i b -
onacci c a s e is s i m p l y D = 5.
PRELIMINARIES
In the f i r s t p l a c e ,
onacci Q u a r t e r l y ,
n-th.
in a c c o r d a n c e with the p r a c t i c e of the F i b -
I h e r e u s e the s y m b o l s
and
L
to denote the
F i b o n a c c i and Lucas n u m b e r r e s p e c t i v e l y ; in o t h s r p a p e r s I
u s e the m o r e widely accepted,
[3].
F
Throughout the following
if l e s s logical, notation u
n, m, k will denote i n t e g e r s ,
n e c e s s a r i l y positive, and r will denote a n o n - n e g a t i v e i n t e g e r .
w h e r e v e r it o c c u r s ,
not
Also,
k will denote an even i n t e g e r , not divisible by 3.
We shall then r e q u i r e
the following
formulae,
elementary
(1)
x
'
and v
2F
, = F L + F L
m+n
m—n - n ~ - m
109
all of which a r e
HO
SQUARE FIBONACCI NUMBERS, E T C .
(2)
2 L , = 5 F F + L L
m+n
m n
m n
(3)
L0 ' = L2 + (-I)™"1 2
2m
m
v
(4)
(F-
, L.
) = 2
(5)
( F , L ) = 1 if
(6)
2 L
1
(7)
3>n
if a n d o n l y if
. J
m
3 L
'
J
F_n=(~-Dn-1Fn
(9)
v
'
L
v
< 12 >
F
(13)
L
,
= - L
m + 20k1 " " . . m
a+2k= m +
t
n
Lk=3.(mod4)
L
\
= (-l)nL
-n
(10)
3 m
'
if a n d o n l y if m = 2 ( m o d 4)
m
(8)
(11)
V. '
April
1 2 ^
L
F
if
2'|k,
3 ^
( m o d L. )
.
k
m
^ °
d
L
k>
^m°d8)
m
THE MAIN T H E O R E M S
T h e o r e m 1.
If
L
n
= x
*
then
n = 1 or 3.
Proof.
If n
i s e v e n , (3) g i v e s
= y2 ± 2 / x2
L
n
7
.
If 115 1 ( m o d 4 ) , t h e n L , =• 1, w h e r e a s if n & 1 w e c a n w r i t e n = 1 +
r
2* 3 # k w h e r e k h a s t h e r e q u i r e d p r o p e r t i e s , a n d t h e n o b t a i n b y (11)
L
and so
2
Ln^x^
since
=. - Li = - 1 ( m o d L, )
- l i s
a n o n - r e s i d u e of
L^. by ( 1 0 ) .
Finally,
1964
SQUARE FIBONACCI NUMBERS, E T C .
Ill
-?
if 113 3 (mod 4) then n = 3 gives L, = 2 , w h e r e a s if n / 3, we
w r i t e a s before n = 3 + 2*3 -k and obtain
L n =. -o L
= - 4 (mod Lj-^)
and again
&
L / x .
n
This concludes the proof of T h e o r e m 3.
T h e o r e m 2.
2
If L = 2x , then n = 0 or ±6.
n
Proof.
If n is odd and L is even, then by (6) n = ± 3 (mod 12) and
so, using (13) and (9),.
L
n
= 4 (mod 8)
and so
L ^ 2x .
n
Secondly, if n = 0 (mod 4), then n = 0 gives
r
if n ^ O , n = 2 * 3 »k and so
2L
n
= - 2L
0
2L / y
n
2
•, i . e .
L ^ 2x
n
2
L, -• 2 • 3
Thirdly, if n = 6 (mod 8) then n = 6 gives
if n / 6 ,
whereas
n = 6 4- 2*3 ' k w h e r e now 41 k, 3/fk and so
2L
and again,
a s before
= 2, w h e r e a s
= ~ 4 (mod L, )
k'
2
whence
L
n
= - 2Lx = - 36 (mod L, )
o
k
- 36 is a n o n - r e s i d u e of
L
n
Finally,
L,
using (7) and (10).
Thus
/
^ 2x".
if n =. 2
(mod 8), then by (9)
L
= L
w h e r e now
- n E 6 (mod 8) and so the only a d m i s s i b l e value is -n = 6, i. e,
This concludes the proof of T h e o r e m 2,
11= - 6 .
112
SQUARE FIBONACCI NUMBERS, ETC.
April
Theorem 3.
If
= x 2 , then n = 0,\±1, 2 or 12.
F
Proof.
If n = 1 (mod 4), then n. = 1 gives
r
n = 1 + 2e 3 • k and so
F
whence
n
F, = 1, whereas if n / 1,
= - F, = - 1 (mod L, )
1
k'
F
/• x . If n = 3 (mod 4), then by (8) F
= F
and -n = 1
J
n
"
-n
n
(mod 4) and as before we get only n = - 1 . If n is even, then by (1)
2
F = F
L,
and so, using (4) and (5) we obtain, if F =• x
n
n
i/2n i/2n
2
2
either
3 In, F , = 2y , L
= 2z .
By Theorem 2, the laty
y
'
y2n
i/2n
ter is possible only for i/2n ='0, 6 or - 6 . The first two values also
satisfy the former, while the last must be rejected since it does not.
or
3iri, F , = y , L
= z .
By Theorem 1, the latter
'
Vi*
V2n
is possible only for i/2n = 1 or 3, and again the second value must
be rejected.
This concludes the proof of Theorem 3.
Theorem 4.
If
F
= 2x Z , then n = 0, ±3 or 6.
n
Proof.
If n = 3 (mod 4), then n = 3 gives
F~ = 2, whereas if n ^ 3,
n = 3 + 2" 3 * k and so
2F
and so F
get
B
only
y
n
n
= - 2 F a = -4
3
(mod L, )
k
4 2x . If n = 1 (mod 4) then as before
n = -3.
must have if
F.
If
z
n
is
even,
then
since
F
= F
and we
-n
n
F = F,
L
we
n
i/2n y2n
= 2x
2
= y ,
/2n
2
F
L,
= 2z ; then by Theorems 2 and 3 we
i/2n
i/2n
see that the only value which satisfies both of these is y2n - 0
2
2
or
F
= 2y ,
L
= z ; then by Theorem 1, the second
J
3
i/2n
i/2n
'
either
of these is satisfied only for' i/2n = 1 or 3.
does not satisfy the first equation.
But the former of these
1964
SQUARE FIBONACCI NUMBERS, E T C .
1.13
This concludes the proof of the t h e o r e m .
REFERENCES
1.
M. Wunderlich,
On the n o n - e x i s t e n c e of Fibonacci S q u a r e s ,
Maths, of Computation, 1_7 (1963) p. 455.
2.
J. H. E, Cohn,
On Square Fibonacci N u m b e r s ,
Proc.
Lond.
M a t h s . Soc. 3^9 (1964) to a p p e a r .
3.
G. H. H a r d y and E. M. Wright, Introduction to Theory of Numb e r s , 3rd. Edition, O. U. P . 1954, p. 148 et seq.
XXXXXXXXXXXK< KXXXXXX
I
/
EDITORIAL NOTE
B r o t h e r U. Alfred cheerfully acknowledges the p r i o r i t y of the
e s s e n t i a l method, u s e d in " L u c a s S q u a r e s " in the last i s s u e of the
Fibonacci Quarterly Journal,
r e s t solely with J. H. E. Cohn.
This
was w r i t t e n at the r e q u e s t of the Editor and the unintentional o m i s s i o n
of due c r e d i t r e s t solely with the E d i t o r .
114
April
EXPLORING THE FIBONACCI REPRESENTATION OF INTEGERS
P r o p o s e d by B r o t h e r U. Alfred on page 72, Dec. 1963,
!!
The Fibonac ci Q u a r t e r ly ! f
The completion of the T h e o r e m stated in the a r t i c l e i s :
The Maximum n u m b e r of different Fibonacci ' n u m b e r s r e q u i r e d to
r e p r e s e n t an i n t e g e r N for which [iMj * = P^ is given by -^ L
This is a c o r o l l a r y of the following t h e o r e m .
For
F
< N £. F , , the n u m b e r N can be r e p r e s e n t e d a s a' sum of
n
n+1
F i b o n a c c i m i n i b e r s s the 1 a r g e s t which is F and the s m a l l e s t g r e a t e r
than or equal to F 9 . M o r e o v e r , the sum n e v e r contains two c o n s e c u tive Fibonacci n u m b e r s , 'We t h e r e f o r e have at m o s t the a l t e r n a t i n g
as c l a i m e d .
t e r m s of indices from 2 to n which gives us
The proof of this t h e o r e m depends upon a L e m m a which is a well
¥+']=[!]•
known Fibonacci Identity that
F~ + F , + F , + . , . + F~ = F~ , 1 - 1
Z
4
6
Zn
2n+l
and that P 0 + F r + F„ + . . . + 'B\ , = F n - 1. The proof of the f i r s t
3
D
{
In-1
Zn
p a r t of this is given by induction and the second p a r t is s i m i l a r l y p r o v e d .
Proof,
J
F o r n = 1, we have P 7 = ¥\ - 1
n = 2, we have F ? + F . = F r - 1 which c l e a r l y shows the L e m m a
holds for 11 = 1, 2ff,
Now a s s u m e that it holds for a i l n <. K? w h e r e K is a fixed but u n specified positive i n t e g e r g r e a t e r than or equal to 3.
i . e . P.^ + F , + . . . + F \ _ = F » T , . . - 1, t h e r e f o r e by addition to both '
2
4
ZK
2K+1
sides we have that F n + F" + . . . + F-,,- + F,,,T7.,.., = F~ _ _,v + F O T f ..^ - T"
2
4
2K
ZK+2
2K+1
ZK+Z
= F
- 1
M
2K+3
which i m p l i e s the L e m m a holds for a l l positive n,
Using this Lemm-* which we shall c.?U L e m m a 1, p a H A ion the
f i r s t c a r t whica was last proved, arid p a r t B for the second p a r t with
the odd \ n d i c e s ; we c-»,n now prove the g e n e r a l t h e o r e m that for
F < N . I F ., , we car. r e p r e s e n t N es a s\im o£ at l e a s t a l t e r n a t i n g
rj
n M"
'
.
Fibonacci nurabe L'L- w h e r e the l a r g e s t is F for N < F , - and which
°
a
n+I
t r i v i a i l v is iu;'t F . , iC^eLf when IN - F . .
n+J
nid
Proof. F o r N - i ,' we have 1 - f\«2' aiulior N = 2» we have 2 = F o3s Now
a s s u m e the t h e o r e m t r u e for a l l M <. k» where- k i H a fixed but unspecified
positive i n t e g e r and n is such that F < k 5. P . , , n > 3*
°
n
n+1
—
Continued on Page 134
r
Now if
PARTITION ^NUMLI'AM'^ l\
v irA^$
D'^Uk
i
GeoiMG in i .<* il ,
i ILLL CR
r u l t v " , #,< , n t a / G c .
Netto [1] illustrate 3 a ,,i(~n> ,
n having exactly
\Jf AMPLER PART?TJOMS
1
<"c 5- ?nur aerating ai* partitions of
!
It i-? shown he re m that
f
. n cJgoritliaur form through ase
p m o n i l o i s , .» ' «oi»- . t i c
Netto's procedure can be r e i n ** *
of simpler partitions wnlzh ? ^ h' • * .4 IJJ range (size and number of
members) but otherwise uru _ > ^-!'-d5
limited partitions per sc an ' a *
procedure
of these range-
Mu? of adapting them to Netto's
enumeration proceoura ore din en >J
algorithmic
Properties
w
I
13 Mv-, d >^»
An "Uviouc application of the
;ofn j'ltd tlon
of both lyp^s of
partitions.
2.
LIMITED-RAi ICI", JN * \,t TiJiC I~T?D PAR I IT JONS
Chr y s ia, 1! s [ ?, ] part it ic " \ e 111 l^wf,/, suit? blymodifled. is u r e d
th r oughout.
To u &,
P t -£ J1.
-- "> -, ,
rO
e n u m e r a t i o n o 1 p a r t i ! io "> ^ o T ^ . u
elusive, no partition ba " £ j e. t
each member being not >^-i
I
- c -tiV' i'lL-^ers from
spe» m e s tJtie
11 tc n v in-
i»„ uoi ir^ore than M. m e m b e r s ,
.'an
,
1 .H m ire than q ? .
rio^' = v e i ,
the set rather ib^ii the CPUUJ >du^ ^ :f i ••nge-'imited partition^ is of
immediate inte r e st ir .'hI s pa • >, ^ a* c fo .^rcifyi sff a
x
- •;» :^-> " 3 - / ,
to the enumeratic»n rotation
V is appended
P ^ i ^ n ^ <r.n^ | ^ p l f 4 p J
=^qi , 4q ? ) denotes the 5 ^t oi ,< -i ior. " ' u v ^ g rbe properties of the
enumeration counterpar c,
The existence cu^/'jLioor t« ^
q p , 4 n ^ simultaneous I -y.
Mc; *
are optimum extreme / ? l , e
c
'
£
Brackets [ J except W J ^ I P o r r c
1
customary manner "VVIJILX^ ^ 1 in 1
less
4
than or equal ro in*
Heaslet [3] .
1 * rt )jt.
V=pr
f[
2p2^nr
j "V "ixed ?, , n 0 , q,, q^s there
am
i l iese a re
u< r rcfeicncf s a i e r^c-d in the
IJ i'id„( a i e itie
t* <-} ^ eted.
0 ? t-ai«
^ integer
See U^p^nsky and
116
PARTITION ENUMERATION BY MEANS
(1)
P l opt_
W
If
p,
P2 opt.
ApriL
= " [-(n/q^l .
= [
Vql
]
•
p
l - p l opt.* p l C a n b e c h a n 8 e d t o Pi 0 p t . ' b u t i f P i > P ! o p t . '
cannot be changed. However, if p 7 —P?
can be changed
f > P?
to p ?
, but if p
Lt O p t .
p7
, p„
In generating the partitions,
found first, then the
cannot be changed.
&
tit O p t .
Ct
the
p, -member
partitions are
(p, +l)-member partitions, e t c . , until the p ? -
member partitions are found.
The procedure used herein for the par-
titions of a typical p-member
set is as follows:
A trial "first" partition is formed from
the p m e m b e r s i s equal to or greater than n
to n ? , the partition initiates the set.
p q, ' s .
If the sum of
but less than or equal
If such is not so, the right-
hand member is augmented so that the sum of the p-members
To form new partitions,
creased by one until either it equals
equals
n?
(or both).
in n, .
the right-hand member is successively inq?
or the sum of the p
The next p-member
members
trial partition is found by
adding one to the member second from the right and replacing all
members to the rightwith the new value of the changed member.
The
desired reinitiating partition is found from the sum of the p members, as before.
The right-hand member is successively increased
by one to form new partitions.
When the possibilities of the particular
second member from the right are exhausted,
one is added to the
third member from the right and the process repeated all over again.
Eventually, all p-mernber
ample for
partitions will be accounted for.
PV(>8, <10 |>2, <5 |>2, <7) follows:
2.6
2.7
3,5
3, 6
3,7
4.4
4.5
4, 6
5,5
2,2,4
2,2,5
2,2,6
2, 3, 3
2,3,4
2,3,5
2,4,4
3, 3, 3
3,3,4
2,2,2,2
2,2,2,3
2,2,2,4
2, 2, 3, 3
2,2,2,2,2
An ex-
1964
OF SIMPLER PARTITIONS
3.
117
APPLICATION OF NETTO'S METHOD
Netto [ 1] considers the enumeration
P(n|p|<q)
of the parti-
tions of n having exactly p members with no member greater than
q.
Netto's methodis limited to q>(n+l-p)
tions being p<n and qp>n,
with the existence condi-
simultaneously.
In the terminology of
this paper,
(3)
where
P(n|p|<q)
-
ta = 0 , 1 ,
S [ i ( n - p + 2 - 3 t r 4 t 2 - . . . -pt
~ j*y
.
2)]
,
Inspection of (3) reveals that the
typical term is
n-p+2-w
(4)
in which w is always zero for
ta - 0, always 3 for
ways greater than 3 for all other
t ' s.
t
= 1, and al-
It can be observed that ex-
cept for the zero value of w, each w in the enumeration
P(n|p|<q)
is the sum of the members of each partition included in the set
P V ( > 3 , < n - p | > l , < [ ^ E ] |>3, <p)
(5)
Thus, except for
(6)
p = 1,
P(n|p|<q) =
n-p+2
2
n-p+2-w.
+ 2
i
It should be noted that (5) does not exist for
There are no w.'s
l
(n-p)<3.
under these conditions, and the summation term
of (6) is accordingly zero.
(7)
p = 2, and/or
The special case of p = 1 is
P(n 11 I <q) = 1
118
P A R T I T I O N E N U M E R A T I O N BY M E A N S
Vs w a s s t a t e d f a i b ^ ,
(n
ru^lu^
risscxihtd
t i c u l a x l y a d a p t a o i e tu > g i f a i . ' rnp . r j ' a , s .
April
herein are par-
To ilias end ? t h e a u t h o r
cars s u p p l y -d i i m i t e a u r a b c r *». " I ^ O S U n g u a g e p r o g r a m s a n d t e s t
e x e m p l a r s for e n u m e i d L n ^ h i h l JS
with t h e B u r r o u g h s ZZO d i g i t a l
computer*
ft1] P E R B" 11C D 3
1.
E„ N e t i o , ^£tLfJ£UC h - J P I C J T?J>IJL=I l<' ~K 5 L e i p s i z ,
2..
G, C h r y s t a l ,
T ^ U J O O ! ^ 2LJ±*JeL!S"
Publishing Co.,
3
V
"°'*
2j
1901, pp. 127,
(Reprint)
Mo^ v i K, i ;b^
J . V. U s p e n s k y -oi 1 M A . h ^ r s i 1 * M c G r a w - H i l l B c o k G~, , >-*
I ler^ntaxy
Number
x 1% 1 9 3 9 , p p . 9 4 - 9 9 .
xxxxxxxxxxxxxxxxxxx*
i ^ M I ^ J I P L V : „ 1 1P-JLI^LLM gJ.A._TJ-Q._3
P a g e 44;
On l i n e 4 r e s d "0 ^ k ;„ ^
P a g e 4-9;
On l i n e 8 :>:—id \i\ir
P a g e 80:
In B - 7 l i n e 2 x = 1/4
VV
and
I
J or
!
£<»r "0 £ k < 2 r "
I r»F 1 F
s Tri2 / 4 i
L2
i ^ O ^ i ^ ~ 25
?
'
£ I I 5 ^ ^ i i ^ i l : : M i ^ r L f PV JLf ':_./OPPME1, NO. 4
Reference 4
In H - 2 5
Chelsea
(
H i e i i r i a u t h o r :,- i^v^L^cLVJJN ,
(i, j = 1, 2, 3 , 4 )
Theory,
A NOTE ON WARING7S FORMULA FOR SUMS OF LIKE POWERS OF ROOTS
S.L. BASIN
Sylvania H e c t r o n i c Systems, Mountain View, California
k
k
k
S, = x " + x'~ + . . . Jr x' u m a yJ be e xlp r e s s e d in
k
I
2
n
t e r m s of e l e m e n t a r y s y m m e t r i c functions or in t e r m s of the coefficients
Sums of rp o w e r s
of:
f(x) = (x-x, )(x--x 9 ) . . . (x-x ) = x'"+p,x ' " + . . „ +p^
J.JL
i^i
A,
X
XX
by Newton 1 s f o r m u l a s , u s u a l l y introduced in. a c o u r s e in the t h e o r y of
equations, for e x a m p l e , J . V. Uspensky ll] .
The r e l a t i o n s h i p between Waring\s formula for s u m s of like
p o w e r s of the r o o t s of a q u a d r a t i c and Lucas n u m b e r s is quite obvious
although p e r h a p s a little too s p e c i a l i z e d for L, E. Dickson. \Z] to have
pointed this out in his text, F i r s t C o u r s e in the Theory of Equations.
In o r d e r to obtain an explicit e x p r e s s i o n for
S,
where
k = 1,
k.
2, 3. . . , f i r s t c o n s i d e r the q u a d r a t i c
2
x"
(1)
T
If we denote the r o o t s by a
(2)
px
T
q = 0 .
and & then (1) m a y be r e w r i t t e n a s
x " + px + q = (x~a)(x--/3) ,
After m a k i n g the t r a n s f o r m a t i o n
x = 1/y
and multiplying by y "' we
obtain^
(3)
I + py + qy" = (1-ay){l-/3y)
.
Differentiating both sides of (3) vvHii i e s p e c t to y and dividing
b o t h m e m b e r s of the differentiated equation by the c o r r e s p o n d i n g m e m b e r s of (3) we a r r i v e at
119
120
A N O T E ON WARING'S F O R M U L A F O R SUMS
April
Equations,
k+1
_,_ 2 ,
, k k - 1 , <*
k
-=a+o: y +. . . +a y
+ - —y
J
1-ay
^
1-ay 7
a
(5)
N
'
and
k+1
a r e both obtained f r o m the g e o m e t r i c
(7)
v
'
=
T^—
1-r
k-1
2
. n
series
rJ + 1
1-r
J=0
for e x a m p l e ,
l e t r = a y a n d m u l t i p l y b y a . A d d i t i o n of (5) a n d (6)
r e s u l t s in.,
)
(8
v
'
_«_+T4_=S1+S?y
1 - a y 1-py
1
2'
k+1
+
where
k+1
. . . +S7v y k - 1 + «—i^M+^Iil^l)
k
S, = a
k
+ /?
(1 ~ a y ) ( l - / ? y )
k
In o r d e r t o e x p a n d t h e l e f t - h a n d m e m b e r of (4) u s i n g (7), l e t r = - p y
~qy , t h e n
k_1
(9)
k k
l
—z = & ( - D W q y 2 ) j + ( - p - q y ) I
1+py+qy
1+
Py-qy
E m p l o y i n g the b i n o m i a l t h e o r e m we m a y w r i t e
( P y+qy 2 ) j = 2 ( - f ^ (py) g (qy 2 ) h
1964
O F LIKE POWERS OF ROOTS
121
w h e r e the s u m m a t i o n i s taken o v e r a l l t w o - p a r t p a r t i t i o n s of j , i . e . ,
for a l l g>0 and h > 0
-P" 2 c iy
1+py-qy
(10)
such that g + h = j .
, ^
,
v /
l x g+h+l
Therefore,
(g+h).r
*•
g h g+2h ,
k k
(-p-2qy)(-p-qy) y b ^
1+py+qy
Now the left-hand m e m b e r s of (8) and (10) a r e equal a s shown
by equation (4); t h e r e f o r e we m a y equate coefficients of like p o w e r s of
k-1
y. Specifically, equating coefficients of y
we a r r i v e at
(11)
w h e r e w e have r e p l a c e d
i for g + 1 and j for h in (10). The s u m -
m a t i o n s in (11) now extend over all i>0 and j >0 such that i + 2j = k,
Combining both s u m m a t i o n s in (11) we have,
(12)
sk=kZ(-l)i+J(i^PV
s u m m e d over all i > 0 , j^O such that i + 2j = k.
-1 we have S, = L, , the kth
(13)
L
k
-k
C l e a r l y , for
p - q
Lucas n u m b e r ; and (12) b e c o m e s
T
^
i>0
j>0
i+2j=k
( i + J " 1 ) i~iTjT
122
A NOTE .ON.WAKING'S FORMULA FOR SUMS
April
Equation .(12) of Waring 1 s•• formula, published in 1762, which can
be extended to. include the sum of kth p o w e r s of the r o o t s of
nth
degree polynomials.
The m a i n point to o b s e r v e is not n e c e s s a r i l y the m e a g e r r e s u l t
given by (13) but the fact that i m p l i c i t in the development of W a r i n g ! s
formula lies the generating function (4) for the Lucas n u m b e r s .
REFERENCES
T h e o r y of Equations,
1.
J. V. Uspensky,
2,
L. E. Dickson, Fir_s_t C o u r s e in the T h e o r y of Equations, John
McGraw-Hill,
1948,
Wiley, 1922,
xxxxxxxxxxxxxxxxxxxx
OMISSIONS •
H-3 and H-8 w e r e a l s o solved by John L. Brown, J r . , The P e n n s y l vania State University, State College, Penn.
The solution to H-16 given in the l a s t i s s u e was compounded from
solutions given by L. C a r l i t z and John L9 Brown, J r .
The v a r i t y p i s t
omitted the c r e d i t line,
H-13 was a l s o solved by John H. Halton,
Boulder,
U n i v e r s i t y of Colorado at
Colorado.
H-15 was a l s o solved by L. C a r l i t z , Duke University, D u r h a m , N. C.
ADVANCED PROBLEMS AND SOLUTIONS
Edited by VERNER E. HOGGATT, Jr.
San Jose State College, San Jose, California
Send all communications concerning Advanced Problems and
Solutions to Verner Ea Hoggatt, Jr. , Mathematics Department, San
Jose State College^ San Jose, California.
This department especially
welcomes problems believed to be new or extending old results.
Pro-
posers should submit solutions or other information that will assist
the editor.,
To facilitate their consideration, solutions should be sub-
mitted on separate signed .sheets within two months after publication
of the problems.
H-34
Proposed
by Paul F. Byrd, San Jose State College,
San Jose,
California
D e r i v e the s e r i e s expansions
sj0
9
4_1
s
J „ ( « ) = I, ( ° ) +
(-1)
I m+k
m= 1
(k = 0, I j 2, 3, , . . ) fortheBessel functions
where
H-35
m-k 1
J^,
2m
of all even orders,
L are Lucas numbers and 1 are modified Bessel functions.
n
n
Proposed
by Walter W. Horner, Pittsburgh,
Pa.
Select any nine consecutive terms of the Fibonacci, sequence and
form the magic squa.re
b
U
Generalize.
8U1U6
U
+U
3U5U7
+u
4u9a2 =
8U3U4 + U1U5U9 + U6U7U2
123
124
ADVANCED PROBLEMS AND SOLUTIONS
H-36
Proposed
by J.D.E.
Konhauser,
Consider a r e c t a n g l e R.
State College,
April
Pa.
F r o m the upper right c o r n e r of R r e -
move a r e c t a n g l e S ( s i m i l a r to R and with sides p a r a l l e l to the sides
of R.
D e t e r m i n e the l i n e a r r a t i o K = L 0 / L
if the c e n t r o i d of the
r e m a i n i n g L shaped region is w h e r e the lower left c o r n e r of the r e moved r e c t a n g l e w a s .
H-37
Proposed by H.W. Gould, West Virginia University,
Morganfown, V/est. Va.
Find a t r i a n g l e with sides n + l , n , n-1 having i n t e g r a l a r e a . The
f i r s t two e x a m p l e s a p p e a r to be 3, 4, 5 with a r e a 6; and 13, 14, 15 with
a r e a 84.
H-38
Proposed by R.G. Buschman, Suny, Buffalo,
N.Y.
(See Fibonacci N u m b e r s , Chebyshev P o l y n o m i a l s ,
Generaliza-
tions and Difference Equations Vol. 1, No. 4, Dec. 1963, pp.
1-7.)
Show
(u
, + s( - b ) r u
)/u =
v
n+r
n-r ' n
H-39
X
r
Proposed by Verner E. Hoggatt, Jr., San Jose State College, San Jose,
California
Solve the difference equation in closed f o r m
C
I O
n+Z
= C
,,+C
n+1
n
+ F
~
n+Z
j
where
C, = 1, C = Z, and F is the nth F i b o n a c c i n u m b e r .
1
Z
n
two s e p a r a t e c h a r a c t e r i z a t i o n s of t h e s e n u m b e r s .
H-40
Proposed by Walter Blumberg, New Hyde Park, L.I.,
Let U, V, A and B be i n t e g e r s ,
N.Y.
subject to the following con-
ditions
(i) U > 1 ,
(ii) (U, 3) = 1;
(iii) (A, V ) = l ;
(iv) V= V^- 1 )/ 5 •
Show A U+BV is not a s q u a r e .
Give
1964
ADVANCED PROBLEMS AND SOLUTIONS
125
SOLUTIONS
EXPANSIONS OF BESSEL FUNCTIONS IN TERMS OF
FIBONACCI NUMBERS
P-Z
Proposed
by P.F.
Byrd, San Jose State College,
San Jose,
California.
D e r i v e the s e r i e s expansions
j 0 ( « ) = t (-Dk £ ( . ) - ij +1 («)
F 2k+1 ,
k=0
where
J
and I
a r e B e s s e l functions, with F 9 1
,
being F i b o n a c c i
numbers.
Solution
by the proposer,
P.F.
Byrd, San Jose State College,
Note: This is a c o r r e c t e d v e r s i o n .
San Jose,
California
Initially this r e a d J ( x ) = . . .
which does not m a k e s e n s e .
It is j u s t as e a s y to d e r i v e the m o r e g e n e r a l s e r i e s expansions
of J ? ( a ) , (p=0, 1 , 2 , . . . ) , for B e s s e l functions of all even o r d e r s ,
Zp
and then to obtain the d e s i r e d r e s u l t as a s p e c i a l c a s e upon setting
p = 0.
We m a k e p r i n c i p a l u s e of f o r m u l a s (6. 1) and (6. 5) p r e s e n t e d
in [l] . Since
J ? (a) is an even function, we f i r s t seek a polynomial
expansion of the form
OO
J
2p(a)=k^0
? 2 k
»PM ^k+l(x)
'
w h e r e from equation (6.5) and [1] the coefficients a r e given by
7r
('\£,2k
5
2k, p ^
=
~
J
J 2 (-2i<* C osv) [cos2kv - cos(2k+2)v]
"'2p
0
being the F i b o n a c c i polynomials defined in [ 1 ] ,
with
_. ,,
Zk+1
known (e. g. , see [2]) that
dv ,
Now it is
7T
J
J \ (-2i« cosv) c o s 2 m v dv = 7T J ,
(~i«) J
(-i«) >
2p
p+m
p-m
0
and a l s o that J (iz) = i n I (-z) = ( - i ) n l (z). Hence we e a s i l y obtain
5
2k, P = ( - 1 ) P + k [ W t t ) I k - P ( a ) - W i ( t t ) W i ( 0 ) ]
•
126
ADVANCED PROBLEMS AND SOLUTIONS
Finally,
taking x = 1/2, and thus with <p
April
(1/2) = F . _, we have
the f o r m a l expansions
J
2
(«)=
P
E (-Dk+Prj
(«)I
(*)-I
(«)T
(all F •
k
k=0
' k+P
~P
k + p + l l M k-p+l v JJ ± 2k+L
which in p a r t i c u l a r yield the solution to the p r o p o s e d problem upon
setting p = 0*
REFERENCES
1.
, P. .F. Byrd, Expansion of Analytic Functions in Polynomials
A s s o c i a t e d -with F i b o n a c c i Numbers, The F i b o n a c c i Quarterly,,
Vol. 1, No,. 1,. pp. 16-29.
2.
G. N. Watson, A T r e a t i s e on the T h e o r y of B e s s e l Functions,
Cambridge, 2nd Edition, 1944, p a 151.
Note;
C o r r e c t e d statements, to P - l . '
Verify
that the polynomials
P
1
(x)
satisfy the differential
equation
•'
( l + x 2 , ) y " +' 3xy f - k(k+2)y = 0 ,
R e a d e r s a r e r e q u e s t e d t.o submit solutions to the problems in the above
mentioned r e f e r e n c e
[l]„
CORRECTIONS IN SAME P A P E R
P a g e 19 Replace 2 ] by [ 2 ] in line 1 1 .
3
P a g e 20 In line 5 r e a d (3) a s r e l a t i o n r e f e r r i n g to footnote 3 In view
of . . . .
'
P a g e 21 Read ^ . + 1 # x as y .(x) in (4. 7)
Page 23 In (5. 6) place absolute value b a r s around the quantity a p p r o a c h i n g the limit z e r o .
SYMBOLIC RELATIONS
H-18.
Proposed
by R..G. Buschman, Oregon State University,
"Symbolic relations
For example,
if
11
Corvaiiis,
Oregon
a r e s o m e t i m e s used to e x p r e s s identities.
F n • and L• n denote,
respectively,
.r
J
Fibonacci and
1964'
ADVANCED PROBLEMS AND SOLUTIONS
127
Lucas numbers,- then
<l:+.L)a=L^.
(l+F)n=F2n
are known identities* where = denotes that the exponents on the symbols are to be lowered to subscripts after the expansion is made,
(a) Prove
(L + F ) n = (2F) n .
(b) Evaluate
(L + L) .
(c) Evaluate
(F + F) . .
(d) How can this be suitably generalized?
Solution by the proposer (now at Sunyf Buffalo, N.Y.)
Consider the generating functions:
u/2
2 e ' H sinh{u./5~/2) = /s" £
n^O
u/2
2 e ' cosh(u/5/2) =
F
u /n!
n
,
•
1
z L u "/n.
n:
n=0
From these and the product formula for power series we can write
F L
(a)
/ 5 s F 2 u ' / /nl = 2 e sinh(ui/5)
=¥ i/5 £
£ Tt~"TTT7 u
x /
x v
*
0
ix
n
n=0
i
n
kf
n=0 k=0
(n-k):
'
'
n
/u\
(b)
^T
k n~l
n
^ u ,
,
/r,n
„
n
n
s
2 T T - r T T i u. = 2 e (cosh uy5 +1} = £• — - — — —
u .
v
i
n^O k~0
n=0
_
n F, F ,
o^> L 2 - 2 ^
k n-k
n
^ u ,
/=- , , /=- • _
n
n
y ,
1
(c)
x
£ 7T""T~~"rTT u ~ ^ e ' ( c o s h u i / b - i ) / 5 = £ — — - - — — u
n=0 k-0
'
•
,
.n=Q
Equating coefficients and multiplying by nj then gives
n
(a)
(b)
1
(c).
k=o © F* L - k
=
° r (F + L>n = (2F) '
" F-
n
£
( f ) L, L 1 = 2 n L + 2
, ~ \k/
ic n-k
n
k=u
n
s ( £ ) F k F n _ k = (2 n L n - . 2)/5
k=0
5o/uf/on &y L. Carlitzf Duke University,
' or
or
(L + L) n =' (2L) n + 2 ,
'
(F + F ) n =. (2L) n - 2)/5
Durham, NX.
As noted by Gould (this Quarterly, Vol. 1 (1963), p. 2) we have
128
ADVANCED PROBLEMS A N D SOLUTIONS
ax
e
bx
oo
n
^ x „
n
n n!
n= 0
- e
a-b
^
°1
A
n
x
ni
T
n=0
a =i(l +^,
It f o l l o w s a t o n c e t h a t
c^
,
bx
ax
April
b
= i(l
n
-f5)
2ax
2bx
n=0
so that
(L + F ) n = 2 n F
(a)
.
n
Similarly
„
£
x /r . T . n ,
2ax
—p ( L + L) = e
2ax
= e
. - (a+b)x , 2bx
+ 2ev
' + e
. 2bx . 0 x
+ e
+ 2e
,
so that
(L + L ) n = 2 n L
(b)
+ 2 ,
while
/
» \2
( « - 5)
so that
TT x .
.n
2ax
2bx
x
s - r r ( F + F) = e
+ e
- 2e
,
n=0 n '
5(F + F ) n = 2nL
(c)
'
- 2 .
n
consider
v
To g e n e r a l i z e t h e s e f o r m u l a s
, ax , bxxr
(e
+ e ) =
r
_
X
,r
(r-s)ax+sbx
( Jx e x
s=0
r
1
,r x s x . ( r - 2 s ) a x . ( r - 2 s ) b x .
;
j 2 (s) e
(e v
+ ev
)
s=0
,
2
Therefore
r
" V
s=0
°~
e
n
n=0
.
.n
nl
n
n *
n
1964
A D V A N C E D . F L O ; - «jy.i>/ir. A N D S O L U T I O N S
/ T J_T J
JLT
/*
li
,
,.
'-
,
5
--.
' ^
-
^
.k
n-k
129
T
^
r
k
r.,
T
7
>k
n-k
In p a r t i c u l a r
(L+L-l-J,)'"1 -- ^ L
SimilarLy-j
J3L,
since
5 r (r+... +F)n - i r (C \
. w h e r e the n u m b e r of P ! - »«= /
r
n
1
5r(F+. . . +F)n = T
'
^ i"i)sdr)(2r"2s)ksn~k .
j* H
, ^ ,
.
-S ?, y-fl
k n—k
T( \
)(2r-2s +l ) K s n K ,
L-d
w h e r e the n u m b e r of r 1 : JC ~ i 1
JU p a r t i c u l a r
A f o r m u l a for
,L +
s e
, -f L + F + . . . T F)
c a n be o b t a i n e d b u t i t i s ^ c r y c o m p l i c a t e d .
W h e n the n u m b e r of L ' s
f
i s e q u a l to the n u m b e r <yc F ;s i\' ic> l e s s c o m p l i c a t e d .
(L+L+F+F) 1 " 1 =r. i ; 4 n 1^ - 2 n + 1 )
In p a r t i c u l a r
.
Indeed s i n c e
/ a x , bx.r, ax
bx.r
,' 2 a x
2bx,r
(e
+ e ) ie
~ e ) = (e
- e
)
it f o l l o w s that
In p a r t i c u l a r
(L
+ L + L + F ! ^ + F
K
-= l ~- l (J F
"
D
n
- 3F, ) .
Z n ••
130
ADVANCED PROBLEMS AND SOLUTIONS
THE
H-1 9
Proposed
by Charles
April
RACE
R. Wall, Texas Christian
University,
Ft. Worth,
Texas
In the t r i a n g l e below [drawn for the c a s e ( 1 , 1 , 3 ) ] , the t r i s e c t o r s of angle, B, divide side, AC, into s e g m e n t s of length F , F
F
10.
n+3
,,
Find:
(i) lime
n-> oo
(ii) lim if
A
v\_
_/v_
n+3
"n+l
Solution
F
by Michael
Goldberg,
Washington,
D.C.
As n ^ o o , the r a t i o F , / F
a p p r o a c h e s t = (-v/5~+ l ) / 2 , a n d
, Q / E a p p r o a c h e s t ~ 2t + 1< Hence, the limiting t r i a n g l e ABC
can be drawn by taking points D and E on AC so that AD = 1, DE = t
3
and EC = t = 2t + 1. Since BD is a b i s e c t o r of angle ABE, the point
B m u s t lie on the c i r c l e which is the locus of points whose d i s t a n c e s
to A and E a r e in the r a t i o A D / D E = l / t . The c i r c l e p a s s e s through
D. If the d i a m e t e r of the c i r c l e is 2r = x + 1, then x / ( x + 1 + t) = l / t
from which
t/(t - 1) = t,2 = t + 1
1
Similarly,
BE is a b i s e c t o r of the angle DBG.
The point B m u s t lie
on a c i r c l e which is the locus of points whose d i s t a n c e s from D and C
a r e in the r a t i o D E / E C = t / t = l / t . If the d i a m e t e r of the c i r c l e is
2
2
2r^ = y + t, then y / ( y + t + t ) = l / t from which
1964
ADVANCED PROBLEMS AND SOLUTIONS
Hence,
131
cos < B A E = - t / 2 ( t + 1) = - {^5~- l ) / 4 and < B A E = 108° .
F r o m w h i c h 20 = 90° - 1 0 8 ° / 2 = 3 6 ° ; 6 = 1 8 ° , ^ = 180 ° - 108° - 3 6 = 1 8 °
Also solved by the proposer
and Raymond Whitney, Penn. State University,
Hazelton,
Penn.
FIBONACCI TO LUCAS
H-20
Proposed
San Jose,
by Werner E. Hoggatt,
California.
Jr., and Charles H. King,
San Jose State
College
If
snow
Q
1
D(e
)= e
0
w h e r e D(A) is the d e t e r m i n a n t of m a t r i x A and L
is the n
n
num be r .
Solution
by John L. Brown, Jr., Penn. StateUniversity,
State College,
R e c a l l that
F
Q
n =
F
F
n
so that (by definition)
2
.Q
F
n+1
nk+1
k.!
k=0
.
n-1
2
k=0
nk
^i
0s5
car
V
F
i ^n
-
k=0
n
n
k,
^
^
i ^n
F
nk-1
k=0—jr
Penn.
th
Lucas
A D V A N C E D P R O B L E M S AND SOLUTIONS
132
April
it i s w e l l - k n o w n t h a t
oo
J? ,nk
•v
—••
k=0 ~'
,
k
x
a x
b x
e
-e
^5
—•
—
[ e . g . e q u a t i o n ( 2 . 1 1 ) , p. 5 of G o u l d ' s p a p e r in V o l . 1, N o . 2, A p r i l
1963j , w h e r e
a
1_ +_V E_ T
=
and
,
1 - VT
Similarly,
L
,
• ,
n
n
u
nk • k
a x .
b x
k,
x = e
+ e
k=0
for the L u c a s n u m b e r s .
But
L , = F , , , + F , ,; therefore,
nk
nk+1
nk-1
°^
y
kt0
F
.
1 ,1 + F .
nk+1
nk-1
kl
, a
(e
2
nk+3.
k=0 "~k!
(1)
n
n
= e
,n
. b
+ e )
un
. b
+ e .
a
" • ^ -
2
k=0
Fnk+1 = F ^ + F ^
and
S
or
1
k!
i
•*-
00
Since
•
jr
00
^
-^-
= S
^ _
V^
k=0
from above, we a l s o have
00
/?\
F
^
k=0
1
, ,,
00
2bEZ± - -^
xv
k!'
k=0
k=0
F
,
nk
k
AVe
°
00
,
k=0
k=0
F
.
,
a
nk-1 _
x
*"'
e
e*o
b
- e
Y5~
,
k=0
S o l v i n g (1) a n d (2) s i m u l t a n e o u s l y , w e find
a
XT
nk+1
k!
(3)
k=0
1
+
( / * / )
e
(4)
k=0
Now,
D(e
Q-n
"nk^ 1 _ 1_
K
*
e
^
00
2
k=0
F ,
a
n
u
. b
+ e
n
/ °° F , ,
n
HK+1 \ I 2
nk-1
kl
M k=0" kl
. n
b
- e
V^r
and
T
n
a
e
n .
T
b
n
- e
i^
\
k=0
nk
kl
F
,
,
nk-1
k!
'
1964
ADVANCED PROBLEMS AND SOLUTIONS
e
a
n ltn
a +b
= e
H-21
n
,n \
+ eb >
/ a
I e
"
e
b
v
L
n
n
'
yr
9
.
T
since L
Proposed by Francis D. Parker, University
of Alaska, College,
,n
b
a
e
N
2
- e
f5
n, u n . .
= a +b for n > 0.
n
—
FIBONACCI PROBABILITY
= e
133
,
q. e„ d.
_±__
Alaska
th
F i n d the p r o b a b i l i t y , a s n a p p r o a c h e s infinity, t h a t t h e n" x F i b o n a c c i n u m b e r , F ( n ) , i s d i v i s i b l e by a n o t h e r F i b o n a c c i n u m b e r {/ F ,
orF2).
Solution by proposer
We u s e f r e q u e n t l y the f a c t that F(n) i s d i v i s i b l e by F(k) if k
d i v i d e s n.
T h e n the p r o b a b i l i t y that F(n) i s d i v i s i b l e by 2 i s 1 / 3 ;
1 2
the p r o b a b i l i t y that F(n) i s d i v i s i b l e by 3 b u t n o t 2 i s (^)(4); t h e p r o b ~~~"
" " 1 2 3
2
a b i l i t y that F(n) i s d i v i s i b l e by 5 but not by 2 or by 3 i s •=• -^ T = ^ | f ;
and i n g e n e r a l the p r o b a b i l i t y that F(n) i s d i v i s i b l e by F(k) but n o t any
F i b o n a c c i n u m b e r of o r d e r l e s s than k i s T T T T ^ - T h e s e p r o b a b i l i t i e s
k(k-l)
a r e a l l i n d e p e n d e n t , s o that the p r o b a b i l i t i e s that F ( n ) i s d i v i s i b l e b y
at l e a s t one F i b o n a c c i n u m b e r of o r d e r not e x c e e d i n g k i s
1
Z
?.
?.
3.2 + T
4 .~ 3: + 5 . 4
' ' ' k(k-l) "
k-2
T h i s s u m i s —-:—, and a s n a p p r o a c h e s infinity, the p r o b a b i l i t y a p K.
p r o a c h e s unity.
A l s o s o l v e d by J. L. B r o w n , J r . , P e n n . State U n i v e r s i t y , , State
College,
Pennsylvania.
134
April
Continued from P a g e 114
k = F , , , we have that k+1 = F ., + F „ , and we a r e through.
If
n+1
n+1
2
k = F ,, - 1, we have k+1 = F , , and we a r e through.
If k = F ,-, - 2
&
n+1
n+1
n+1
we have k+1 = F , - 1, which by L e m m a 1 (A or B) ? can be r e p r e sented as c l a i m e d and we a r e through again. T h e r e f o r e let us cons i d e r k i F ., - 3.
n+i
Now the r e p r e s e n t a t i o n f o r k in this form can best be e x p r e s s e d
a s k = F +a
~ F » +a - F „ +a
. F
. + . . . + a^F- + a 0 F n
n - 2 n-2
n-3 n-3
n-4 n-4
3 3
2 2
w h e r e a. = 0 or 1 for 2 £. i S n - 2 , and a. = 1, i m p l i e s that a. ± 1 = 0.
i
I
x
i
Now t h e r e a r e only two p o s s i b i l i t i e s for a ? and a, in this r e p r e s e n t a tion.
E i t h e r a ? = a . = 0, or a
^ a«.
If the f i r s t c a s e is t r u e for k,
we can r e p r e s e n t k+1 in the r e q u i r e d m a n n e r , simply by adding 1 to k
in the form of a ? = 1. If the second c a s e is t r u e for k, we then c l a i m
that t h e r e e x i s t s at least one place in the r e p r e s e n t a t i o n w h e r e a. ='
a. . , = 0 , since o t h e r w i s e , k = F ., - 1 which we have a l r e a d y taken
l+l
n+1
. '
c a r e of above.
T h e r e f o r e we can r e p r e s e n t k+1 by the following:
k+1 = F +a ~F „ + . . . + a . , - F . ,., + a.' F . , +. . . + a 'F 0 + a 0 F 0 +1
n
n-2 n-2
i+2 i+2
i-l i-l
3 3
2 2
Now c o n s i d e r the e x rp r e s s i o n from a. , n F . ._ on and the r e s u l t i n gto in-'
i+2 1+2
equality.
a i+2
. i 7 F .i+2
. , - - 1 <. F n - 1, - 1,
l7+a,
i - l. F .i - l, + . . . + a 3Q F 3. + a 02F „2 +1 < F i+3
by our Inductive A s s u m p t i o n , Also by the Inductive A s s u m p t i o n , we
can r e p r e s e n t the e x p r e s s i o n from a. , ~ F . ~ on in the p r o p e r form
which i m p l i e s that we can then a l s o r e p r e s e n t k+1 in the p r o p e r f o r m .
This shows that the proof holds for all positive i n t e g e r s N. Q. E. D.
P h i l Lafer, Oak H a r b o r , Ohio
BEGINNERS' CORNER
Edited by DMITRI THORO
San Jose State College
T H E E U C L I D E A N A L G O R I T H M II
1.
INTRODUCTION
In P a r t I [ l ] w e s a w t h a t t h e g r e a t e s t c o m m o n d i v i s o r
numbers
could be c o n v e n i e n t l y c o m p u t e d via the f a m o u s
algorithm.
Suppose that exactly
c o m p u t e t h e g. c. d. of
s
and
n
t
We t h e n h a v e
(1)
s = t q , + r ,
0 < r
< t
(2)
t = r , q^ + r ,
0 < r?
< r
0 < ^
< rz
0 <
< r^
rx = r 2 q
(4)
r2 = r3 q4 + r ^
(5)
r 3 = r 4 q5 + r ^
(n-1)
N
'
r
(n)
x
'
r
0
n-3
= r
3
+ xy
^ q
-,+r
n-2 ^n-1
r
r
l-^
2
r
>r
3
r
3
> r
etc,
135
+ r
2
+
4
3
r 4
+ r
5
x
4
, < r
9
n-1
n-2
.
t > r: + r2
(3')
(5')
0<r
1
n-1,
5
q. > 1, t h e a b o v e e q u a t i o n s i m p l y
(2 1 )
(4')
T^
Q < x
~ = r
, q + 0
n-2
n - 1 ^n
Since each quotient
Euclidean
s t e p s ( d i v i s i o n s ) a r e r e q u i r e d to
(s > t ) .
(3)
of . t w o
136
F'F.CJNr^Rt' ^ ' r i i orw
F r o m (21) and ^ i,
(2 r . + 2 r J + r_,
2 r . , etc.
-„ ~, * -' ,• , ' /, I r o m (4 f ) ?
r
Sim:^"Ly, L - -.. t'i \
-1
Continuing In ch. s JI* u % . .v
of Fibonacci n u m b e r s .
April
2 r
+ r
>
^ -! " r . > (3 r . + 3 r . ) +
j
i<^ * he g e n e r o u s abundance
Thus
t •£• r , + r > 2r -1-r, > 3r n -f2r, > 5r ,+3r r
1 2
2 j ~~
.:>
4 *~
4
5
> ¥ , r _ +F n r _
"~ n-1 n - 2
n-2 n-1
2. A BASIC RESULT
Since the r e m a i n d e r s form a sti'i^tiy d e c r e a s i n g sequence with
r
, the last n o n - z e r o r e m a i n d e r .
n-1
c
„ v r
XI-&
"*
11- I
Consequently,
n-1
xi" L
n-L
) j - ! '""
n- 1
n-Z
n+1
To s u m m a r i z e , if n divisions a r e r e q u i r e d to compute the g. c. d. of
st
s and t, then t is at l e a s t as l a r g e as the (n + 1)
Fibonacci
number!
3. LAME'S THEOREM
Although the Euclidean a l g o r i t h m is over 2,000 y e a r s old, the
following r e s u l t was e s t a b l i s h e d by G a b r i e l Lame in 1844.
Theorem
The n u m b e r of divisions r e q u i r e d to find the g„ c. d. of two n u m b e r s is n e v e r g r e a t e r than five t i m e s the n u m b e r of digits in the s m a l l e r
number.
Proof.
Let
4> designate the golden r a t i o .
*pn = F n <t> + F n ^ | ,
In [2] it was shown that
n = l , 2, 3, . . .
1964
BEGINNERS' CORNER
Now since 2 > <t> = (1 + ^/5)/Z9
2F
+ F
n
137
we see that
, > F
ii-1
4> + F
,
or
n-1
n
> *n
F
n+2
Replacing n by n~l
section yields
and using the "basic r e s u l t " of the p r e c e d i n g
t > *n-1
.
To complete the proof note that
(i)
if t has
(ii)
log t > (n-1)
(iii)
Thus d > ( n - l ) / 5
d digits then d > log t
log 0
log 4> > 1/5 • .
or
n < 5d .
REFERENCES
1.
D. E. Thoro, "The E u c l i d e a n Algorithm I, " F i b o n a c c i Q u a r t e r l y ,
Vol. 2, No. 1, F e b r u a r y 1964.
(Note
that i n e x c e r c i s e s E8 and E10,
v
should be r e p l a c e d byJ
^
respectively. )
2.
D. E. Thoro,
N(F , . ,
n+1
"The Golden Ratio:
tions, " Fibonacci Quarterly,
(F . , , F ) and m a x (n, F - l )
n+1
n
n
F ) and m a x N (n, F - l )
n'
n
Computational
Vol. 1,
pp. 5 3 - 5 9 .
xxxxxxxxxxxxxxxxxxxx
No. 3,
Considera-
October 1963,
EXPLORING FIBONACCI NUMBERS WITH A CALCULATOR
BROTHER U. A L F R E D
St. Mary's College, C a l i f o r n i a
It h a s often been noted that the study of n u m b e r s is both e x p e r i m e n t a l and t h e o r e t i c a l in c h a r a c t e r .
culator,
Even in the days before the c a l -
to say nothing of the c o m p u t e r ,
the t r u l y g r e a t m a t h e m a t i -
cians often a r r i v e d at beautiful r e s u l t s on the b a s i s of o b s e r v a t i o n
and n u m e r i c a l w o r k before they p r o c e e d e d to proof and t h e o r e t i c a l
justification.
If we find o u r s e l v e s enjoying calculation and seeing
tangible r e s u l t s , we a r e in v e r y good company and need not w o r r y
about the attitude of the t h e o r i s t who is afraid to soil his hands with
n u m b e r sa
In this vein,
the following exploration is p r o p o s e d .
It m a y be
o b s e r v e d by looking at a list of F i b o n a c c i n u m b e r s that in c e r t a i n c a s e s
F h a s the n c o r r e s p o n d i n g to the t e r m i n a l digits of the n u m b e r .
n
°
.
°
Thus F 5 is 5; F
is 514229; F 6 l is 2504730781961. As long as
we have a table of Fibonacci n u m b e r s on hand we can p r o c e e d to m a k e
such v e r i f i c a t i o n s .
But suppose we set out to find all t h e s e coinci-
d e n c e s for F i b o n a c c i n u m b e r s up to a c e r t a i n level such a s
F,
n
nrm-
In the a b s e n c e of t h e s e n u m b e r s we now have an i n t e r e s t i n g m a t h e m a t i c a l p r o b l e m involving computation,
One v e r y simple way to p r o c e e d would be to take the s u c c e s s i v e
F i b o n a c c i n u m b e r s modulo 10, 000. In. other w o r d s we would c o n s i d e r
only the last five digits and forget about all those that go before.
This
is a s t r a i g h t f o r w a r d p r o c e d u r e but it would be long and tedious and
subject to e r r o r .
In fact,
once a m i s t a k e in introduced, a l l r e s u l t s
t h e r e a f t e r would be vitiated.
there are several ways.
T h e r e m u s t be a b e t t e r way.
Perhaps
We shall look f o r w a r d to both the n u m e r i c a l
r e s u l t s and the method employed in a r r i v i n g at t h e m .
A d d r e s s ail c o m m u n i c a t i o n s r e g a r d i n g this p r o b l e m to:
U. Alfred, St. M a r y ' s College, California.
in the i s s u e of D e c e m b e r , 1964.
138
Brother
The solution will a p p e a r
AMATEUR INTERESTS IN THE FIBONACCI SERIES-PRIME NUMBERS
JOSEPH MANDELSON
U.S. Army Edgewood Arsenal, Maryland
My i n t e r e s t in the Fibonacci s e r i e s was born in 1959 when it
was noticed that the p r e f e r r e d r a t i o s developed in the r e s e a r c h of m y
colleague, H. E l l n e r ,
and l a t e r included in D e p a r t m e n t of Defense
Handbook HI0 9 [ 1 ] , w e r e 1, 2, 3, 5 and 8.
F r o m r e c o l l e c t i o n of a
brief mention in college a l g e b r a , this was recognized as the f i r s t few
t e r m s of the F i b o n a c c i .
To t e s t the supposition that the p r e f e r r e d
r a t i o s would all be from this s e r i e s , the next one was calculated and,
s u r e enough, it was 13.
Then it was noted that the s a m p l e s i z e s ,
Acceptable Quality Levels (AQL's) and lot size r a n g e s of all sampling
s t a n d a r d s since Dodge and Rornig [2] w e r e s e r i e s a p p r o x i m a t e l y of
the type:
(1)
u
^
n+^
~- u
+u
n+1
n
In fact the l a t e s t v e r s i o n of M i l i t a r y Standard Mil Std 105 [3] shows
s a m p l e s i z e s which a r e a l m o s t exactly the Fibonacci s e r i e s itself.
These o c c u r r e n c e s w e r e too r e m a r k a b l e to be a s c r i b e d to m e r e c o incidence and my i n t e r e s t led m e to examine the s e r i e s e m p i r i c a l l y .
A c c o r d i n g to Dickson [4], the l i t e r a t u r e on this subject is r i c h , extending a s it does from the y e a r 1202 to the p r e s e n t .
However, it is
a l m o s t completely unavailable to m e and, I suspect, to m o s t o t h e r s .
On developing the s e r i e s
u
from
n = 0 to n - 25 or so, in-
spection soon r e v e a l e d that two t h i r d s of the s e r i e s c o m p r i s e d odd
n u m b e r s and exactly e v e r y t h i r d u
was even.
It did not take much
to a s c e r t a i n why this is so. In this way I found that n, the ordinal of
u
in the s e r i e s w a s , in a m a n n e r of speaking, the d e t e r m i n a n t of the
p r o p e r t i e s of u .
Thus, if z is a factor of u
a factor of u ? , u„ , e t c .
it will infallibly be
T h e r e f o r e , in general* if n is c o m p o s i t e ,
so is u n x(except
^ for the c a s e n• = 4. un = 3),
' but if n is ^p r i m e , un
m a y be p r i m e . My f i r s t g u e s s that, since the density of odd n u m b e r s
139
140
AMATEUR INTERESTS IN THE FIBONACCI
in the Fibonacci is twice that of the even numbers,
April
the density of
primes would be greater than in the cardinal number domain was proven wrong when the primality of u
prime.
was found dependent on n being
The next supposition of equal density was shown to be wrong
when u~, = 1346269 was found to be a composite of 557 and Z417.
When u^ 7
and u . ,
were also determined to be composite it became
obvious that the density of primes in u was less than that of the
3
r
n
cardinal domain.
Several other interesting details were elucidated after extending
and examining the series,
finally to n = 130.
first down to n = 50 then to n = 100 and
No u
is divisible by n except when n ~ 5 or
powers of 5. For example uR = 5 and \i?i- - 75025. Except for U/,
every u
seem>s to have at least one prime factor which has not been
J
n
a factor of any previous
factors.
u ; some have two or three such new prime
Surely, any theory of prime numbers might profit from Fib-
onacci considerations.
However, the first gain from the extension of study of the series to n = 100 was a remarkable regularity found from the fact that
if P
is a prime factor of u
it will also factor, more generally,
to
J
n
^
n
u.
where j goes from 1 to 00. Consider the multiple j and let this
be expressed as a sum of multiples of powers of P , reduced to a
minimum of terms, and provided that no multiples of the powers of
P
n
> P .
n
Thus:
(2)
x
'
j = aP° + b P 1 + c P 2 + . . . qP1"
n
n
n
^ ii
J
where a, b, c . . . q may be zero but must always be less than P .
Then u.
will be divisible by
P x + 1 where P
is the lowest xpower
J
jn
n
n
term of P
in the sum of multiples of powers of P = i.
n
r
r
II J
Example 1.
The first prime to divide u
is 2 (P = 2) and it divides the
r
N
n
n
'
third number
we have:
(n = 3) in the series: U3 = 2.
From the above lemma
1964
SERIES -- PRIME ]NUMBERS
S u m of P
Ordinal
*
p
n
x
jn
J
terms = j
3
1
Pn
0
6
2
P1
n
1
^0
9
P
24
8
P3
30
10
33
11
^1
Pn°
P
3
^3
+ Pn
n
^3
P n J_
+ p + Pn
r.0
n
p
is divisible
by P x
J
0 n
+
P
nl +
P
n
0
+ Pn
n
n
Since P
_L
jn
X
0
3
u.
n
3
+1
P
0
P°
n
u.
1
1
= Pn = 2
1
= P
2
2
= 4
n
8
+ 1 _ PX = 2
n
+ 1 P 4 , 16
=
n1 + 1
1
141
+1
=
=
n
P
n2
34
46368
= 4
332040
PX = 2
3524578
n
n
= 2, no m u l t i p l e s other than 0 or 1 a p p e a r in the sum of
p o w e r s of P
= j . Actually the sum of m u l t i p l e s of power t e r m s for
j = 11 should read:
11 = I P
0
1
2
^
E x a m p l e 2.
Another p r i m e dividing u
tioned,
5
n
j terms = j
1
P°
P
n
X
0
n
10
2
2P°*
n
0
20
4
4P°n
0
25
5 P1
30
n
6 P°
n
35
1
^
is 5 (P
0
+2
1
+ 2
3
7
1
+ P1
2P°n
50
10 2 P
25
25 P
2
n
1
n
+
n
P1
n
0
0
1
2
= 1 + 2 + 8.
= 5) and, a s a l r e a d y m e n -
it divides the fifth n u m b e r in the s e r i e s : u_ = 5.
mcike a table:
Ordinal
Sum of P
Jn
0
+ I P + OP + I P = P + P + P = 2
n '
n
n
n
n
n
n
Again we
u. is divisible
jn
by P_ x+1
J
n
u.
p 0 + l
= P1 = 5
5
n
n
1
p o + i -P
55
= 5
n
n
p0+l ^P1
6765
= 5
n
n
1+1
75025
P
= P 2 = 25
n
n
P o+i ^ P 1 = 5
832040
n
n
p 0 + l = PX
9227465
= 5
n
n
1+1
12586269025
P
= P 2 = 25
n3
n 2+1
P
= P = 125
n 59425114757512643212875125
n
142
A M A T E U R I N T E R E S T S IN T H E F I B O N A C C I
*
April
0
T h e m u l t i p l e Z of 2 P
in this c a s e ) is u s e d .
At a l a t e r t i m e ,
recommended
p l a y s no parts
o n l y t h e p o w e r of
in a p r i v a t e c o m m u n i c a t i o n *
D i c k s o n [4] a s a r e f e r e n c e
a c c o r d i n g to
Dickson,
to t h e l i t e r a t u r e .
the above w a s
T h e o r e m V of e i g h t i n t h e f o l l o w i n g
"If
n
(zero
D r . S« M .
I d i s c o v e r e d that t h e s e findings w e r e known to L u c a s [ 5 ] .
ticular,
P
stated by
Ulam
In t h i s
In
par-
Lucas
as
form:
i s t h e r a n k of t h e f i r s t t e r m
u
containing the p r i m e
n
&
ris the f i r s t t e r m divisible by
f a c t o r p to t h e p o w e r A, t h e n u
X+l
X+2
^ >n
p
a n d n o t by p
; t h i s i s c a l l e d t h e l a w of r e p e t i t i o n of p r i m e s
in t h e r e c u r r i n g to; s e r i e s of u . "
n
On r e a d i n g t h i s it i s c l e a r t h a t p r e c e d e n c e i n t h i s f i n d i n g l a y
with Lucas who had,
ically.
moreover,
s t a t e d it m o r e c l e a r l y and e c o n o m -
F a r from being discouraged, however, I continued m y s e a r c h ,
l i s t i n g all p r i m e n u m b e r s up to 10009 and l a b o r i o u s l y t e s t i n g the p r i m a l i t y of m o s t
!
u
s
u p to
n = 130.
Of c o u r s e ,
p r i m e s up to 10009
a r e s u f f i c i e n t o n l y to t e s t u
u p to n = 40 d i r e c t l y b u t t h e f a c t t h a t
if z d i v i d e s u
it w i l l d i v i d e u .
h e l p e d ag r e a t l yJ . N e v e r t h e l e s s i t
^
jii
n
•
s p e e d i l y b e c a m e a p p a r e n t t h a t r e p e a t e d d i v i s i o n of u
g r e a t e r than
u . j - on a d e s k c a l c u l a t o r w a s n o t o n l y l a b o r i o u s b u t i n c r e a s i n g l y p r o n e
t o e r r o r a s t h e n u m b e r of d i g i t s in u
r o s e a b o v e 1 0 . If o n l y t h e r e
J
°
n
w e r e s o m e w a y to e l i m i n a t e s o m e of t h e t r i a l d i v i s i o n s !
A s t u d y7 of 1t h e p r i m e s , P , w h i c h d i v i d e u
r e v e a l e d t h a t t h eJy
n
n
w e r e a l l of t h e f o r m
(3)
P n = an + 1
Since
P
i s p r i m e it i s o b v i o u s t h a t a n h a d to b e e v e n s o t h a t a n ± 1
n
could be odd,
T h e r e f o r e e i t h e r a o r n o r b o t h h a d to be e v e n .
C l o s e r s t u d y of t h e p r i m e s i n d i c a t e d t h a t , w h e n
e v e n , it w a s a l w a y s n e c e s s a r y to a d d o n e to
a
and
n
even,
P
= a n + 1,
never
a n - 1.
an
a
and
n w e r e both
to g e t
P , i . e. w i t h
I c a n n o t e x p l a i n t h i s but,
e m p i r i c a l l y , i t t u r n s o u t t h i s w a y . Now i t w a s p o s s i b l e t o c u t d o w n o n
t h e n u m b e r of d i v i s i o n s r e q u i r e d to d e t e r m i n e t h e P
which would
•L
n
1964
SERIES — PRIME NUMBERS
divide u n .
Thus:
a.
Calculate
b.
D e t e r m i n e whether
c.
Divide u
d0
If u
2n + 1 (If n is even, d e t e r m i n e only 2n + 1).
2n + 1 a n d / o r
2n - 1 a r e p r i m e .
by any p r i m e n u m b e r d e t e r m i n e d in a and b.
is not divided in c. calculate
n
Repeat setps c. and d,
e„
3n + 1.
f. If u
is not divided in step e, calculate
n
even, d e t e r m i n e 4n + 1 only).
e.
&
143
4n + 1
If n is
Continue until the P
which divides u is found.
n
n
The r e l a t i o n s h i p found above m a y be e x p r e s s e d a s follows;
If P is any p r i m e t h e r e e x i s t s an n such that P
an - 1 will divide u
b e r > 0. )•
without r e m a i n d e r (a being s o m e whole n u m -
The only exception is
/
J
= an + 1 or
1
P
=5
which divides u c = 5.
5
n
It is p o s s i b l e that the above r e l a t i o n s h i p would r e p a y i n v e s t i g a tion in p r i m e n u m b e r t h e o r y . In the past, a n u m b e r of f o r m u l a s have
been p r o p o s e d for the p u r p o s e of generating p r i m e n u m b e r s . In e v e r y
c a s e the f o r m u l a s have been found faulty in one or m o r e of the following r e s p e c t s :
a.
The density of p r i m e s gene r a t ed lias been much lower than
the t r u e density of p r i m e s .
b.
They have g e n e r a t e d c o m p o s i t e n u m b e r s ,
c.
They have r a r e l y been capable of g e n e r a t i n g p a i r e d p r i m e s
(two consecutive p r i m e s which differ by 2, e. g'. 11 and 13).
The formula given in (3) suffers only in generating v e r y m a n y c o m p o s i t e s . However, the p r o c e d u r e c l e a r l y furnishes a c r i t e r i o n w h e r e by (empirically) it h a s been found that,
divided by P
if n is p r i m e ,
u
will be
only when p , determined, as in (3), is p r i m e .
If
this can be proven, new light m a y b e shed by the proof on this a g e - o l d
problem.
144
AMATEUR INTERESTS IN THE FIBONACCI
April
REFERENCES
1.
D e p a r t m e n t of Defense Handbook HI09 "Statistical P r o c e d u r e s
for D e t e r m i n i n g Validity of S u p p l i e r s ' A t t r i b u t e s Inspection, "
6 May I960.
2.
Dodge and Romig, "Sampling Inspection Tables, " Second E d i tion, 1959, John Wiley and Sons, Inc.
3.
M i l i t a r y Standard Mil Std 105D,
"Sampling P r o c e d u r e s and
Tables for Inspection by A t t r i b u t e s , " in p r o c e s s of publication
by D e p a r t m e n t of Defense.
Revision Mil Std 105C is dated 18
July 1961.
4.
Dickson,
" H i s t o r y of the Theory of N u m b e r s , " Chapter XVII,
5.
R e c u r r i n g S e r i e s , Lucas 1 u , v .
°
-n
n
Lucas, "Sur la the'orie d e s n o m b r e s p r e m i e r s , " Atti R. Accad.
Sc. Torino (Math. ), 11, 1875-6, 928-937, cited in [ 4 ] .
XXXXXXXXXKXXXXXXXXXX
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P l e a s e notify the Managing E d i t o r AT ONCE of any a d d r e s s change.
The P o s t Office D e p a r t m e n t , r a t h e r than forwarding m a g a z i n e s m a i l e d
t h i r d c l a s s , sends them d i r e c t l y to the d e a d - l e t t e r office.
Unless the
a d d r e s s e e specifically r e q u e s t s the F i b o n a c c i Q u a r t e r l y be f o r w a r d e d
at f i r s t c l a s s r a t e s to the new a d d r e s s , he will not r e c e i v e it.
(This
will u s u a l l y cost about 30 cents for f i r s t - c l a s s p o s t a g e . ) If p o s s i b l e ,
p l e a s e notify us AT LEAST
dates:
THREE WEEKS PRIOR to publication
F e b r u a r y 15, A p r i l 15, October 15, and D e c e m b e r 15.
RENEW YOUR SUBSCRIPTION! I
A MOTIVATION FOR CONTINUED FRACTIONS
A.P. HILLMAN and G.L. ALEXANDERSON
University of Santa Clara, Santa Clara, California
This Q u a r t e r l y is devoted to the studyof p r o p e r t i e s of i n t e g e r s ,
e s p e c i a l l y to the study of r e c u r r e n t s e q u e n c e s of i n t e g e r s .
We show
below how such s e q u e n c e s and continued fractions a r i s e n a t u r a l l y in
the p r o b l e m of a p p r o x i m a t i n g an i r r a t i o n a l n u m b e r to any d e s i r e d
c l o s e n e s s by r a t i o n a l n u m b e r s .
We begin with the equation
x2 - x - 1 = 0 .
(1)
One can e a s i l y see that t h e r e is a negative root between -1 and 0 and
2
a p o s i t i v e root between 1 and 2, for e x a m p l e by graphing y = x - x - 1.
We call the positive root r .
This n u m b e r has been known since a n -
tiquity a s the "golden m e a n . " We now look for a sequence of r a t i o n a l
a p p r o x i m a t i o n s to
r.
A r a t i o n a l n u m b e r is of the form
(and q ^ 0).
p / q with p and q i n t e g e r s
We t h e r e f o r e wish two s e q u e n c e s
p
l*
q
r
P
2'
P
3S ' * '
(2)
q s
z
q s
3
of i n t e g e r s such that the quotients p / q
get a r b i t r a r i l y c l o s e to r .
a r e a p p r o x i m a t i o n s which
It would a l s o be helpful if each new a p -
p r o x i m a t i o n w e r e obtainable s i m p l y from p r e v i o u s o n e s .
We go back to equation (1) and r e w r i t e it as
(3)
\
x = 1 +1 .
J
x
This s t a t e s that if we r e p l a c e x by r
(4)
x
'
1 + x
145
in
146
A MOTIVATION FOR CONTINUED FRACTIONS
the result Is r
April
and suggests that if we replace x in (4) by an approx-
imation to r we will get another approximation.
We now change (3)
into the form
and consider
of l / x ,
x,
to be an approximation to r.
is the same as that of x,
tive e r r o r of x ?
(ic e. j
and? if x,
1 + 1 /x ) is lower than that of x , ,
adding 1 increases the number but not the e r r o r .
x?
in (5) is a better approximation to r
since
It can be shown that
that x , if x
We now let our first approximation x
Pi Ah
The relative e r r o r
is positive, the rela-
> 0.
be a rational number
and substitute this in (5) obtaining
*i
Pi + ^
= 1 + ——;r— = 1 + -— =
x
^pi/qr
*
We therefore choose p ?
to be p, + q,
our third approximation is
general, the
(n + l ) - s t
p
p /q„
p
i
and q ? to be p, .
r
w ith p
=p
approximation p
(6)
Pn+l
= P
n
<7>
VU
= P
n•
It follows from (7) that q
=p
(8)
%
'
+ q
r
,,
and q„ = p . In
has
, ; substituting this in (6) gives
.
is between 1 and 2 we use 1 as the first approximation,
i. e. , we let p. = q, = 1.
from (8) that p
, /q
-f q
Similarly,
n
p ,, = p + p
^n+1
n
^n-l
Since
.
i
This means that p ? = 2 and it now follows
is the Fibonacci number
F ,1 -
Then (7) implies
that q = F
and we see. that the sequence of quotients F , , / F of
^•n
n
T.
A
n+1 n
consecutive Fibonacci numbers furnishes the desired approximations
to the root r of (1). It can be shown that this sequence converges to
r in the calculus sense.
1964
way.
A MOTIVATION FOR CONTINUED FRACTIONS
We next c o n s i d e r the p r o b l e m of a p p r o x i m a t i n g
The n u m b e r s is the positive root of
147
s = vTO in this
x 2 - 10 = 0 .
(9)
We w r i t e (9) in the f o r m s
2
9 =1
(x - 3)(x + 3) = 1
(10)
( x - 3) = l / ( x + 3)
x = 3 + l / ( x + 3)
and change (10) into
l
*u-=3+
n+1
3 -I- x
(11)
Again,, if x
x
is a positive a p p r o x i m a t i o n to s s it can be seen that
, is an a p p r o x i m a t i o n with s m a l l e r r e l a t i v e e r r o r .
sequence of r a t i o n a l a p p r o x i m a t i o n s
p ... = 3p + lOq ,
^n+1
^n
^n
p /q
T h e r e is a
with
q 1 = p + 3q
^n+1 *n
^n
Letting the f i r s t a p p r o x i m a t i o n be 3 5 i. e. , letting p, = 3 and q
we obtain the sequence
3 / 1 , 1 9 / 6 , 117/37, ... .
which can be shown to converge to s0
Equation (11) contains the equations
1
X
2
= 3
+
3 + xx '
1
x
3
+
~3 + x 2
Substituting the f i r s t of t h e s e into the second gives us
= 1,
148
A MOTIVATION FOR CONTINUED FRACTIONS
*3
= 3
+
April
*1
6+3 +
X;L
If this is substituted into x 4 = 3 + 1/(3 + x ) and if we let x,
be 3,
we obtain
l
x, = 3 +
In this way we
the x .
n
can write continued fraction expressions for any one of
Then it is natural to let the infinite continued fraction
3+-
1
6+
represent the limit
l
6 +. ..
s of the sequences
x
defined by (11) and
The infinite continued fraction for the root
r
ofx
X]
= 3.
- x - 1 - 0
1+
i
+
1
1 +...
whose elegant simplicity is worthy of the title "golden mean. "
xxxxxxxxxxxxxxxxxxxx
FIBONACCI AND LUCAS NUMBER TABLES
Those interested may secure bound mimeographed tables of the
first 1505 Fibonacci numbers, F , and the first 1506 Lucas numbers,
L , by sending two dollars to Professor Jack K. Ward, Westminster
College, Fulton, Missouri.
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900
MAXEY BROOKE
Sweeny, Texas
In 1202, a r e m a r k a b l e m a n w r o t e a r e m a r k a b l e book.
The
m a n w a s L e o n a r d o of P i s a , known a s Fibonacci, a b r i l l i a n t m a n in an
intellectual wilderness.
The book Liber Abacci (The Book of the
Abacus) introduced A r a b i c n u m b e r s into E u r o p e .
In the book w a s a s e e m i n g l y s i m p l e little p r o b l e m :
"A p a i r of r a b b i t s a r e enclosed on. all s i d e s by a wall.
To find
out how m a n y p a i r s of r a b b i t s will be born in the c o u r s e of one y e a r ,
it being a s s u m e d that e v e r y month a p a i r of r a b b i t s will p r o d u c e
another pair,
and that r a b b i t s begin to b e a r young two months after
t h e i r own b i r t h . "
On the m a r g i n of the m a n u s c r i p t , F i b o n a c c i gives the tabulation:
A pair
1
First
2
Second
3
Third
5
Fourth
8
Fifth
13
Sixth
21
Seventh
34
Eighth
55
Ninth
89
Tenth
144
Eleventh
233
Twelfth
377
He s u m s up h i s calculations
149
150
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 A p r i l
". - . w e see how we a r r i v e at it.
We add to the f i r s t n u m b e r
the second one* i . e M 1 and 2; the second to the third; the t h i r d to the
fourth; the fourth to the fifth; and in this way, one after a n o t h e r , until
we add together the tenth and eleventh and obtain the total n u m b e r of
r a b b i t s — 377; and i t i s p o s s i b l e to do this in this o r d e r for an infinite
n u m b e r of m o n t h s . "
T h e r e the m a t t e r lay for 400 y e a r s .
[ 1 ] of a s t r o n o m y fame,
21, . . .
In 1611, Johann Kepler
a r r i v e d at the s e r i e s 1, 1, 2, 3, 5, 8, 13,
T h e r e is no indication that he had a c c e s s to one of F i b o n a c c i ' s
h a n d - w r i t t e n books (The Liber Abacci was not published until 1857 [ 2 ] ) .
At any r a t e , in d i s c u s s i n g the Golden Section and phyiiotaxis, Kepler
wrote:
" F o r we will always have a s 5 is to 8 so is 8 to-13, p r a c t i c a l l y ,
and a s 8 is to 13, so is 13 to 21 a l m o s t . I think that the s e m i n a l f a c culty i s developed in a way analogous to this p r o p o r t i o n which p e r p e t u a t e s itself, and. so in the flower is displayed a pentagonal standard^
so to speak.
I let p a s s all other c o n s i d e r a t i o n s which might be a d -
duced by the m o s t delightful study to e s t a b l i s h this t r u t h .
,f
Simon Stevens (1548-1620) a l s o wrote on the Golden Section.
The editor of his w o r k s , A. G e r a r d [3] a r r i v e d at the formula for
e x p r e s s i n g the s e r i e s in 1634
Un+£
, o = Un+1
,, + Un
A hundred y e a r s m u s t p a s s before the p r o b l e m is again considered.
In 1753,
R. Simpson [4] d e r i v e d a formula,
implied by
Kepler
u
, u ,, - u 2 = (-i) n + 1
n-1 n+1
n
'
A s e c o n d h u n d r e d y e a r s p a s s by and the s e r i e s again c o m e s under
study.
In 1843, J. P . M. Binet [5] d e r i v e s an a n a l y t i c a l function for
d e t e r m i n i n g the value of any F i b o n a c c i n u m b e r
2 n ^ 5 U a = (l + ^ / 5 ) n - ( I -
sf5)n
1964 FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 151
The following y e a r , B. Lame [6] f i r s t used the s e r i e s to solve
a p r o b l e m in T h e o r y of N u m b e r s . He investigated the n u m b e r of ope r a t i o n s needed to find the GCD of two i n t e g e r s (it does not exceed 5
t i m e s the n u m b e r of digits in the s m a l l e r n u m b e r ) .
Two y e a r s l a t e r , E. Catalan [7.] derived the i m p o r t a n t formula
n-1
L
_n
un -
T
5 n ( n - l ) ( n - 2 ) , 5 2 n ( n - l )(n-2)(n- 3)(n-4) J
+ - - _ - - _ _ +_^™ T ___ r i T ____„ + . . .
By now, the s e r i e s had r e c e i v e d enough attention to d e s e r v e a
name,
It was v a r i o u s l y called the Braun S e r i e s , the S c h i m p e r - B r a u n
series^ the Lame s e r i e s and the G e r h a r d t s e r i e s .
A. B r a u n [8] , applied
s c a l e s of pine c o n e s ,
to the a r r a n g e m e n t of the
Schimper i s completely unknown.
a l r e a d y been mentioned,
B e r n a r d Lami,
the s e r i e s
Lame''has
but the n a m e h a s been c r e d i t e d to F a t h e r
a c o n t e m p o r a r y of Newton and the d i s c o v e r e r of the
p a r a l l e l o g r a m of f o r c e s .
G e r h a r d t i s probably a m i s - s p e l l i n g of
Girard.
Edouard
Lucas [ 2 0 ] , who dominated
the field
of r e c u r s i v e
s e r i e s during the p e r i o d 1876-1891 , first applied F i b o n a c c i ' s n a m e
to the s e r i e s and it has been known a s Fibonacci s e r i e s since then.
About this t i m e ,
1858, Sam Loyd claimed to have invented the
checkerboard paradox [9].
j o u r n a l in 1868 [ 1 0 ] .
It i s f i r s t found in p r i n t in a G e r m a n
Today i t s e e m s p r o p e r to call it the C a r r o l l
P a r a d o x after Lewis C a r r o l l [11] ( C h a r l e s Dodgson, 1 832-93) who was
quite fond of it.
Before the c e n t u r y ended, a n u m b e r of f a m i l i a r r e l a t i o n s w e r e
found.
Among them:
(V i s the n t h Lucas n u m b e r . )
1876, E , Lucas [12]
u 2 , + u 2 - u 7 .,
n-1
v"
4n
n
Zn-t-1
= vt: - 2
2n
152
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 A p r i l
V
4n+2 = U L + 1
+ 2
u n + p = u n ~ p (u+i) p
u n " p = u n (u-i) p
1886, E. Catalan [13]
[14]
2
n+2 P
Un +,1l - p Un+l+p
. . . - Un+1
" Up
+. = ( - l )
U2 - U
U x = (-l)n"p+1 U .
n
n - p n+p
n-1
1899, E. Landau [15] r e l a t e d the s e r i e s
"i
n=l
( 1'/ U 92n)
to L a m b e r t ' s s e r i e s and
n
to the theta s e r i e s .
~^
A complete list can be found in Vol. 1 of D i c k s o n ' s " H i s t o r y of
the Theory of N u m b e r s . "
This s m a l l h i s t o r y ends a r b i t r a r l y at 1900 for the p r e g m a t i c
r e a s o n that a m e r e listing of twentieth c e n t u r y developments would fill
a m o d e r a t e l y sized volume. It would be i n t e r e s t i n g to see F i b o n a c c i ' s
r e a c t i o n to the application of his rabbit p r o b l e m to such d i v e r s e s u b j e c t s a s m u s i c a l composition [16], p r o c e s s optimization [ 1 7 ] , e l e c t r i c a l network t h e o r y [ 1 8 ] , and genetics [19].
1.
2.
REFERENCES
J. Kepler "De nive s e x a n g u l a " 1611.
B. Concompagni, II Liber Abbaci de Leonardo P i s a n o Pubiicato
da B a l d a s s u r e Boncompagni, Roma MCDDDLVTL
3.
A. G e r a r d .
" L e s O e u v r e s Mathematique de Simon Stevin" Leyde
1634, pp. 169-70.
4.
R. Simpson, "An explanation of an o b s c u r e p a s s a g e in A l b r e c t h
G i r a r d ' s c o m m e n t a r y upon Simon Stevin's w o r k s " P h i l . T r a n s .
Roy. Soc. (London) 48, I, 368-77 (1753).
FIBONACCI NUMBERS; THEIR HISTORY THROUGH 1900
J. P. M. Binet " M e m o i r e sur ['integration des equations
153
lineaires
a u x d i f f e r e n c e s finies d'u,i o r d r e quelconque, a coefficients v a r i a b l e s " Comp. Ren. Accad. Sci. P a r i s 17, 563 (1843).
B. L a m e .
"Note s u r la limite du n o m b r e des divisions dans la
r e c h e r c h e du plus grand commun d i v i s e u r e n t r e deux n o m b r e s
e n t i e r s " Compt. Rend. Acad. Sci. 19, 1867-70 (1944).
E.
Catalan.
"Manual des candidates a Ecole Poly t e c h n i q u e "
tome 1, P a r i s 1857, p. 86.
A. B r a u n .
" V e r g l e i c h e n d e Untersuchung liber die Ordnurng d e r
Schuppen an den Tannenzapfen als Einleitung zur Untersuchung
der
BlattersteHung
uberhaupt.
" Nova
Acta
Acad.
Caes.
Leopoldina 15, 199-401 (1830).
M. G a r d n e r " M a t h e m a t i c s ,
Magic, and M y s t e r y " Dover, New
York, 1956, p . 133.
Zeitschrift fur M a t h e m a t i k und Physik, 13, 162 (1868).
Lewis C a r r o l l " D i v e r s i o n s and D i g r e s s i o n s " Dover, New York,
p. 316-7 (1956).
E. Lucas "Note sur la t r i a n g l e a r i t h m e t i q u e de P a s c a l et sur la
s e r i e de L a m e " Nouvelle C o r r . Math. 2, 74 (1876).
E. Catalan.
Melanges M a t h e m a t i q u e s Vol. 2, Liege 1887, p .
319.
E. Catalan, Mem. Soc. Roy. Sci. Lieges (2), 13, 319-21 (1886).
E. Landau. "Sul la s e r i e des i n v e r s e s des n o m b r e s de F i b o n a c c i
Bull. Soc. Math. F r a n c e 27, 298-300 (1899).
L. Dowling and A. Shaw.
"The Schillinger System of Musical
Composition" C a r l F i s c h e r , New York, 1946.
S, M. Johnson.
"Best Exploration for Maximum is F i b o n a c c i a n "
Rand Corp. R e s e a r c h Memo. RM-1590 (1955).
A. M. M o r g a n - V o y c e .
" L a d d e r - N e t w o r k Analysis using F i b -
onacci N u m b e r s " P r o c . IRE Sept. 1959, pp. 321-2.
F . E. Binet and R. T. Leslie "The Coefficients of inbreeding in
c a s e of Repeated F u l l - S i b - M a t i n g s " J. of Genetics, June I960,
pp. 127-30.
E. Lucas " T h e o r i e des Fonctions Nume'riques Simplement P e r i o d i q u e s " A m . J. Math. 1, 1 8 4 - 2 2 0 ( 1 8 7 8 ) .
ELEMENTARY PROBLEMS AND SOLUTIONS
Edited by A . P . H I L L M A N
University of Santa Clara, Santa Clara, California
Send all c o m m u n i c a t i o n s r e g a r d i n g E l e m e n t a r y P r o b l e m s and
Solutions to P r o f e s s o r
A. P . Hillman,
Mathematics
U n i v e r s i t y of Santa C l a r a , Santa C l a r a , California.
Department,
We w e l c o m e any
p r o b l e m s believed to be new in the a r e a of r e c u r r e n t sequences a s
well a s new a p p r o a c h e s to existing p r o b l e m s .
The p r o p o s e r should
s u b m i t his p r o b l e m with solution in legible form, p r e f e r a b l y typed in
double spacing, with n a m e ( s )
and a d d r e s s of the p r o p o s e r
clearly
indicated.
Solutions to p r o b l e m s listed below should be submitted within
two months of publication.
B-38
Proposed
by Roseanna
Torretto,
University
of Santa Clara, Santa Clara,
C h a r a c t e r i z e s i m p l y all the sequences
n+2 ~
B-39
Proposed
by John Allen
Fuchs,
n+1
University
c
California
satisfying
n
of Santa Clara,
Santa Clara,
California
Let F , = F . = 1 and F ^ = F ,. + F for n > 1 .
1
2
n+2
n+1
n
P r o v e that
F ^ < 2 n for
n+2
B-40
Proposed
If H
n
by Charles
R. Wal/, Texas Christian
n > 3 .
University,
Fort Worth,
Texas
is the n - t h t e r m of the g e n e r a l i z e d F i b o n a c c i s e q u e n c e ,
°
i.e.,
154
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
R
1
= pr , H
2
155
= pr + q , H ± = H ± 1 + H
forn>l,
^
n+2
n+1
n
show that
2 kH. = x(n + 1)H
q .
;
J - - H , , + 2p + H
, 2-,
.*
n+2
n+4
^
k=l
B-41
Proposed
by David L. Silverman,
Beverly
Hills,
California
Do t h e r e e x i s t four d i s t i n c t positive F i b o n a c c i n u m b e r s in a r i t h metic progression?
B-42
Proposed
by S.L. Basin,
Sylvania
E x p r e s s the (n + l ) - s t
of F .
n
B-43
Proposed
Electronic
Systems, Mountain
Fibonacci n u m b e r
View,
F ,,
as a function
Also solve the s a m e p r o b l e m for Lucas n u m b e r s .
by Charles
R. Wo//, Texas Christian
Fort Worf/i, Texas
University,
(a) Let x n > 0 and define a sequence x,
k > 0S w h e r e f(x) = y l + x.
by x,
Find the l i m i t of x,
(b) Solve the s a m e p r o b l e m for
f(x) =
v 1 + 2x
(c) Solve the s a m e p r o b l e m for f(x) =
V2 + 3x
(d)
California
, = f(x, ) for
as k —• ©o.
Generalize.
SOLUTIONS
FIBONACCI AND PASCAL AGAIN
B-16
Proposed
by Mar/or/e Bicknell,
Terry Brennan,
Show that if
Lockheed
San Jose State College,
Missiles
San Jose,
and Space Co., Sunnyvale,
California,
California
and
156
April
ELEMENTARY PROBLEMS AND SOLUTIONS
then
2F
n-1
Rx"
F
,F
n-1 n
,F
n-1 n
F
- F
n+1
2F
F
n-1
F
n
F
4-1
n n+1
nFn+l
n+1
( T h e r e a r e s o m e m i s p r i n t s in the originaL s t a t e m e n t . )
Solution hy\L. Carlitz, Duke University,
Durham, N.C.
Put
R
a m a t r i x of o r d e r
k + 1; for example
0
R
l
(r, s = 0, 1, . . . , k) ,
(.'.)
k=
1
R
=
- 1
"0
0
1"]
0
1
1
_1
2
1J
2 "
1_
It is e a s i l y verified that
F
(1)
R;
.
n-1
F
n
( n = 1, 2,
...)
n+1
Indeed this is obviously t r u e for
n = 1.
A s s u m i n g that the formula
holds for n, we have
n+1
R"
n-1
n+1
n+1
1
1
L
n+1
In the next place we notice that the t r a n s f o r m a t i o n
n+2
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
x' =
!
157
y
y1 = x + y
induces the t r a n s f o r m a t i o n s
,2
:'
=
T2:
x'3
.1
=x
+ 2xy + y
=
,2 ,
x 1 y1 =
, ,2
x'y1 =
y'
and so on.
xy + y 2
^ x'y« =
y
T
2
y
y
Z
3
+ y
2
2 , 3
x y + 2xy + y
'
= x
+ 3x y + 3xy
A l s o it is evident from (1) that T,
Tn.
1
4
xy
+ y
is given by
\ x ^ = F ,x.+ F y7
/
n-1
n
) y ( n ) = F n~
X +'
F
n+ly
We t h e r e f o r e get
(x x ( n ) ) 2
'
= F 2 , x 2 + 2 F . F xyJ + F 2Jy 2
n-1
n-1 n
n
2
„n ) x (n) (a) = F
T-:
j x ' Vs
= Fn-1
n_lFnx
( n )2 2
(wyM)
)
'
+
(F 2
+
F^F^Jxy +
2
2
,= FF 22x X2 2+ +2 F— F ,. xy
- • + ~F~
,, yn
n n+1 J
n+1 J
A/so solved by the proposers.
LAMBDA FUNCTION OF A MATRIX
B-24
Proposed by Brother U. Alfred, St. Mary's College, California
It is evident that the d e t e r m i n a n t
F^y 2
158
E L E M E N T A R Y P R O B L E M S AND SOLUTIONS
h a s a v a l u e of z e r o .
n+1.
n+2
n+1
n+2
n+3
n+2
n+3
'n+4
P r o v e t h a t if t h e s a m e q u a n t i t y
k is added to
t h e v a l u e b e c o m e s (-1) n - 1 k.
e a c h e l e m e n t of t h e a b o v e d e t e r m i n a n t ,
Solution
April
by Raymond Whitney, Pennsylvania
State University,
Hazelton
Campus
U s i n g& t h e b a s i c F i b o n a c c i r e c u r s i o n f o r m u l a F
e l e m e n t a r y row and c o l u m n t r a n s f o r m a t i o n s
,, + F a n d
in=F
n+2
n+1
n
we m a y r e d u c e the d e -
t e r m i n a n t to:
n+1
F
n+1
F
n+2
-1
-1
0
which is
(-1)
k
= k(F F
n n+2 • F n + 1 >
•
1
by a b a s i c i d e n t i t y .
Also solved by Mor/or/e Bicknell,
who p o i n t e d out the
San Jose State College, San Jose,
r e l a t i o n to
"Fibonacci
California
Matrices
and
Lambda
F u n c t i o n s , " b y M, B i c k n e l l a n d V. E . H o g g a t t , J r . , t h i s Q u a r t e r l y ,
V o l . 1, N o . 2; R. M . G r a s s l ,
F.D. Parker, State University
U n i v e r s i t y of S a n t a C l a r a ,
California;
of New York, Buffalo, N.W., R.N. Vawter, St. Mary's
College,
California; H.L. Walton, Yorktown H.S., Arlington, Virginia; and the proposer.
EXPONENTIALS OF FIBONACCI NUMBERS
B-25
Proposed by Brother U. Alfred, St. Mary's College,
Find an e x p r e s s i o n for the
T n = 1, T 1 = a,
T 9 = a,
...
In-I
2n
2n-2
California
general term(s)
of t h e
where
and
"2n+l
,T7 T9
.
Zn 2 n - 1
.
sequence
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
159
Solution by Vassili Daiev, Sea Cliff, LA., N.Y.
The f i r s t few t e r m s a r e
F
F
0
It is e a s y1 to see that
n
k = F,(n+3)/2
IO \ /o ^
F
2 , 1
T
n
is odd.
Also solved by J.AM.
= a
F
F
3
F
2
4
w h e r e k = F , '/ o x if n is even and
(n/2)
Hunter, Toronto,
Ontario,
Canada
who suggested the c o n s i d e r a t i o n of log T ;
Ralph Vawter, St. M a r y ' s College, California, and the p r o p o s e r .
E d i t o r i a l Comment: The p r o b l e m can be solved by showing that
log T L . = log T
+ log T ,
&
mL4
° m+2
° m
MAXIMIZING A DETERMINANT
B-28
Proposed by Brother U. Alfred, St. Mary's College, California
Using the nine F i b o n a c c i n u m b e r s F ? to F , ~ (1, 2, 3, 5, 8, 13,
21,
34, 55),
d e t e r m i n e a t h i r d - o r d e r d e t e r m i n a n t having e a c h of
t h e s e n u m b e r s a s e l e m e n t s so that the value of the d e t e r m i n a n t is a
maximum0
Solution by Mar/one Bicknell,
San Jose State College,
California
By c o n s i d e r i n g combinations of F i b o n a c c i n u m b e r s which give
m i n i m u m and m a x i m u m values to s u m s of the form
abc + def + ghi,
the following d e t e r m i n a n t s e e m s to have the m a x i m u m value obtainable with the nine F i b o n a c c i n u m b e r s given:
F .10 F4 *7
F
F10F9F3 + F?F6F5 + F4F3F2 - ( F ^ F g + F ^ F ,
F6
F9 F3
F2
F 5 F8
+F
39796 "
38300.
Also solved
by the
proposer.
1496
8 F 4 F 6>
160
B-29
E L E M E N T A R Y P R O B L E M S AND SOLUTIONS
Proposed by A.P. Boblett,
U.S. Naval Ordnance Laboratory,
April
Corona, California
Define a g e n e r a l F i b o n a c c i sequence such that
F , = a;
1
F 0 = b;
2
F
F
n
n
=F
n -02
+ F
.,
n-1
= F , - - F ,. ,
n+2
n+1
Also define a c h a r a c t e r i s t i c n u m b e r ,
n > 3
n < 0
"~
C> for this sequence, w h e r e
C = (a + b)(a - b) + a b .
Prove:
F , . F , - F 2 = ( - l ) n C , for a l l n .
n+1 n - 1
n
Solution by P.O. Parker, State University
From
of New York, Buffalo,
N.Y.
F(n) = F(n - 1) + F(n - 2), F ( l ) = a, F(2) = b,
we get
_,
b - a r n . b - as n
F(n)x =
-j s + — — j r ,
1 +s
1 +r
where
r and s a r e solutions of the q u a d r a t i c x
2
- x - 1 = 0.
Using
the fact that r + s = - r s = 1, d i r e c t calculation yields
F(n + l ) F ( n - 1) - F 2 ( n ) = [(a - b)(a + b) + ab ] ( - l ) n .
The well known r e s u l t
F(n + l ) F ( n - 1) - F 2 ( n ) = ( - l ) n is the
s p e c i a l c a s e in which a = b = 1.
Also solved by Mor/or/e Bicknell,
San Jose State College, San Jose,
Donna J. Seaman, Sylvania Co.; R.N. Vawter, St. Mary's College,
the proposer, J.A.H. Hunter, of Toronto,
Ontario.
California;
California; and
BROTHER U. A L F R E D
St. Mary's College, California
In the October* 1963 issue of this journal [l] , the author discussed some of the periodic properties of Fibonacci summations.
It
was noted that a certain determinant was basic to these considerations.
Its main characteristics and value were indicated and a pro-
mise was given of additional explanation in some later issue of the
Fibonacci Quarterly.
The purpose of this article is to set forth the
manner of evaluating these determinants on an empirical basis.
The
proof of the general validity of the results obtained is to be found in
an article by Terry Brennan in this issue of the Quarterly [2],
To fix ideas the determinant of the sixth order will be used.
Written in a form that brings out its
Fibonacci characteristics it
would be:
3 3
11
1214
l l
2214
2 I
4 2
3 2"
3 3
21
3 3
3 2
3224
325
5
5 3
5432
3 3
5 3
5Z34
53
5 6 - :L
855
8452
3 3
3 5
8254
8 55
136+86-]L
5
13 8
13482
3 3
13 8
13284
16
Z
6
3
6
5
6
86
; iV
+
1 6 - ]L
+
l
6
-] L
251
2412
+
2
6
- ]L
352
+
3
6
-]
I
+
151
5
13 8
5
5
5
A c e r t a i n subtlety should be noted i n the f i r s t line a s the f i r s t " 1 "
stands for F - and the second f o r F , .
By s e p a r a t i n g t h e t e r m s
of t h e f i r s t
column into groups, t h e
p r o b l e m c a n b e c h a n g e d t o t h a t of e v a l u a t i n g t h r e e d e t e r m i n a n t s w i t h
f i r s t c o l u m n s a s i n d i c a t e d be low:
81
82
SOME DETERMINANTS INVOLVING POWERS
(1)
(2)
1
1
26
1
April
(3)
13
D e t e r m i n a n t s (1) and (2) can be evaluated in t e r m s of what shall be
called the BASIC POWER DETERMINANT.
D e t e r m i n a n t (3) will be
developed in t e r m s of the c o f a c t o r s of the f i r s t column which involve
the basic power d e t e r m i n a n t m i n u s one of its r o w s .
BASIC POWER DETERMINANT
The f i r s t d e t e r m i n a n t h a s a common factor in each of its r o w s
(the f a c t o r s a r e 1 , 2 , 3 , 5 , 8 , 1 3 r e s p e c t i v e l y ) . If t h e s e f a c t o r s be taken
out of the d e t e r m i n a n t ,
determinant.
1
1
1
1
5
24
23
22
2
5
342
4
5 3
4
8 5
4
13 8
3322
3223
3
5332
5233
5 • 3"
8352
8253
8
13 3 8 2
13 2 8 3
1
13 • 8'
5
5
13
we have what will be called the b a s i c power
F o r the sixth o r d e r , it is a s shown below:
5
This d e t e r m i n a n t is a s p e c i a l c a s e of the m o r e g e n e r a l d e t e r m i n a n t
in which the f i r s t row s t a r t s with any F i b o n a c c i n u m b e r w h a t s o e v e r
1964
OF FIBONACCI NUMBERS
4
F F. ,
F5
F?F?,
i l-l
1
5
F
* i+1
5
F
F3+1F2
F
i+1 i
1+1 1
4
F F
i+5 i+4
F
i+5
3 2
F F
i+5 i+4
4
F.
1 1-1
i l-l
4
83
i» 1
i
«*t
F2 F 3
i+1 i
F.
2
3
F F
i+5 i+4
4
F F
i+5 i+4
F5
1-
1
F5
1
5
F
i+4
The basis for evaluating this determinant is the r e l a t i o n
F F
- F
F
= (-l)n+1F
n n+k+1
n+1 n+k K }
*k
To evaluate the determinant we proceed to p r o d u c e z e r o s in the f i r s t
row.
This i s done by multiplying the f i r s t column by F . , and sub-
tracting from this F . times the second column; then multiplying the
second column by F . , and subtracting from this
column; etc.
F . t i m e s the t h i r d
The operations for the f i r s t and second columns would
be as follows:
4
F
i
r+i<Fi-iFi+i
- w
f+2<Fi-lFi+2
-
F
.4
F
and so on.
/T_
^
i+3<Fi-lFi+3'.-
F F
'"
\Wijr2Ft+2..
i i+l)
^w
F F
4
= t-1) F i F i+i •...
i i+2)
=
;
^
lA
i„ ^4
"
i+3
It is clear that the second row would have a factor
the third a factor
F~, etc.
F, ,
Thus, after eliminating the common fac-
tors and expanding by the non-zero t e r m in the last column of the
row, the absolute value of the resulting d e t e r m i n a n t would, be F , F?
•F^F.Fp. multiplied by the basic power d e t e r m i n a n t of the fifth o r d e r .
If we adopt the notation
An
84
SOME DETERMINANTS INVOLVING POWERS
ApriL
to r e p r e s e n t the b a s i c power d e t e r m i n a n t of the nth o r d e r , this r e s u l t
m a y be e x p r e s s e d a s
l*61 = n
•
(F4)
|A |
.
1=1
In g e n e r a l , for a d e t e r m i n a n t of o r d e r
l^nlvn
1
n,
(F.)-
|Aa
| .
1=1
Since the p r o c e s s m a y be r e p e a t e d , it is not difficult to a r r i v e at the
final r e s u l t :
A
=
n
Fn-X .
In the p a r t i c u l a r c a s e of o r d e r six,
It is i n t e r e s t i n g to note that the values of t h e s e b a s i c power d e t e r m i n a n t s a r e independent of w h e r e we s t a r t in the F i b o n a c c i s e q u e n c e .
SIGN OF THE BASIC POWER DETERMINANT
It is i m p o r t a n t to be able to d e t e r m i n e the sign of the b a s i c d e t e r m i n a n t value i n a s m u c h a s we shall combine the values of d e t e r m i n a n t s ( l ) a n d (2) with the values of the c o f a c t o r s of d e t e r m i n a n t (3).
The c o n s i d e r a t i o n s involved a r e a bit t e d i o u s .
c a s e s a c c o r d i n g a s n i s of the form
We d i s t i n g u i s h four
4k, 4k+l, 4k+2, or 4k+3.
The
following t h r e e f a c t o r s d e t e r m i n e the outcome:
(i)
The sign introduced by expanding from the l a s t e l e m e n t in the
f i r s t row;
(ii)
The signs of the t e r m s , of the d e t e r m i n a n t r e s u l t i n g after e a c h
of the s t e p s indicated above.
plus or a l l m i n u s .
These t e r m s will be e i t h e r a l l
1964
(iii)
O F FIBONACCI NUMBERS
The sign of
A
85
which is the s e c o n d - o r d e r d e t e r m i n a n t of the
f i r s t p o w e r s of the Fibonacci n u m b e r s in the Last two r o w s .
Thus, for the sixth o r d e r d e t e r m i n a n t we have been c o n s i d e r ing,
&z
is
The final outcome is a s follows:
(i)
(ii)
F o r o r d e r 4k or 4k+l, the sign is always plus;
F o r o r d e r 4k+2 or 4k+3, the sign a g r e e s with that of A
As noted p r e v i o u s l y ,
the b a s i c power d e t e r m i n a n t enables us
to evaluate d e t e r m i n a n t s (1) and (2).
The l a t t e r can be brought to
this form by shifting the f i r s t column so that it b e c o m e s the l a s t
column.
BASIC POWER DETERMINANTS WITH ONE ROW MISSING
To evaluate the t h i r d d e t e r m i n a n t we find the cofactors of the
e l e m e n t s in the f i r s t column.
F o r the e l e m e n t in the f i r s t row, this
cofactor is a b a s i c power d e t e r m i n a n t after r e m o v i n g c o m m o n fact o r s , but for a l l the o t h e r s it is e s s e n t i a l l y a b a s i c power d e t e r m i n a n t
with one row m i s s i n g .
order
The absolute value of such a d e t e r m i n a n t of
n with a m i s sing row between the k and (k+l)st
row will be
r e p r e s e n t e d by
A
(k I k + 1)
n
'
the i m p l i c a t i o n being that the absolute value d o e s , not depend on the
p a r t i c u l a r F i b o n a c c i n u m b e r with which it s t a r t s .
When developing
such a d e t e r m i n a n t the p r o c e d u r e is the s a m e a s for the development
of the b a s i c power d e t e r m i n a n t , only in this c a s e t h e r e is a gap.
calculation for a d e t e r m i n a n t of o r d e r
The
n with a row m i s s i n g between
the t h i r d and fourth rows can be s u m m a r i z e d s c h e m a t i c a l l y in the
following m a n n e r .
The column headings a r e Fibonacci n u m b e r s .
A
table e n t r y is the power to which the F i b o n a c c i n u m b e r at the head of
the column i s being r a i s e d .
plied t o g e t h e r .
The quantities in any one row a r e m u l t i -
In the f i r s t row we have the r e s u l t of the f i r s t step in
86
SOME DETERMINANTS INVOLVING POWERS
April
the evaluation in which the order is changed from 9 to 3; in the s e c ond, the f a c t o r s r e s u l t i n g in reducing the determinant from order 8
to o r d e r 7; e t c ,
1
1
0
1
1
1
1
1
0
1
0
1
1
1
1
0
1
1
1
1
1
0
1 .
0
1
1
1
1
0
1
1
1
1
1
1
0
1
1
0
The sum of the quantities in any column gives the power of the F i b onacci n u m b e r in the determinant.
In the above case
A9(3 | 4) = FJF^FJFJFJFJFJF, .
The s a m e r e s u l t would have been obtained if the gap had been after
the sixth row.
In g e n e r a l ,
if the determinant is of order n, a gap
after the kth . row or the (n~k)th row gives the same result.
The p a t t e r n o b s e r v e d is as follows: (1) A reduction of 2 in the
p o w e r s of F-. to F
inclusive (if k is l e s s than n-k); (2) A r e duction of 1 from F, M to F , inclusive; (3) No reduction therex
k+1
n-k
'
after. If n-k is l e s s than k, the roles of k and n-k are r e v e r s e d .
Finally,
if n - k equals k (even n), there would be a reduction of 2
from 1 to k and no r e d u c t i o n thereafter.
These r e s u l t s m a y be summarized in the following formulas.
FORMULA FOR k LESS THAN n-k
A n (k ' k+1)
k
i=l
FORMULA FOR n - k
A n*(k
k+1]
. , n-k
n F^1"1 n
1
i=k+l
Fn'\
1
LESS THAN k
k
n x l
n
i=l
F.
X
n
i=n-k+l
.
n
.,,
F n " 1+1
n
i=n-k+l
n
i=k+l
x
Ff-i+1
1964
O F FIBONACCI NUMBERS
87
FORMULA FOR k EQUAL TO n - k
n/2
A (n/2 | n/2+1)
n
,n-i-l
F
i=l
n
XI
. n,,
Fn-i+l
i
T h e s e f o r m u l a s a r e not difficult of application.
However,
for the
sake of convenience (in view of future c o n s i d e r a t i o n s ) and a s a p o s sible guide to r e a d e r s the r e s u l t s for o r d e r s 12 and 13 a r e set down
in d e t a i l .
Since, h o w e v e r ,
t h e r e is s y m m e t r y in k and n - k
only
the f i r s t half need be given in each c a s e .
TABLE OF AT 0 (v k
12
£
k+1)
F
8
x
1
2
1
10
10
4
3
1
2
10
9
4
3
1
3
1
10
11
3
10
9
4
4
10
9
4
4
1
5
10
9
5
4
1
6
10
9
5
4
1
12
TABLE OF A 1 3 ( k | k+1)
k
1
F
F,
Fn
11
10
2
10
10
3
10
4
1.0
5
10
6
10
9
9
9
9
11
10
12
x
7
6
5
4
3
2
1
1
8
7
6
5
4
3
2 •
2
1
8
7
6
5
4
3
3
2
1
8
8
7
6
5
4
4
3
2
1
8
7
7
6
5
5
4
3
2
1
8
7
6
6
6
5
4
3
2
1
9
9
9
8
13
SIGN OF POWER DETERMINANT WITH ONE LINE MISSING
The c o n s i d e r a t i o n s leading to the d e t e r m i n a t i o n of the sign of
power d e t e r m i n a n t s with a line m i s s i n g a r e involved.
i s p r e c i s e l y the s a m e a s for the power d e t e r m i n a n t .
The a p p r o a c h
The r e s u l t s for
88
SOME DETERMINANTS INVOLVING POWERS
a l l values of n,
April
i ( s u b s c r i p t of the leading F i b o n a c c i n u m b e r in the
d e t e r m i n a n t ) and k (as defined for the b r e a k point but taken modulo
4) a r e listed in the following t a b l e .
k
4r
4r+l
1
-
+
2
-
-
3
+
4
+
4r+2
i odd
4r+2
i even
4r+3
i odd
+
+
-
+
-
-
+
+
'+
+
-
+
~,
4r+3
i even
-
+
EVALUATION O F THE ORIGINAL DETERMINANT
We noted p r e v i o u s l y that the o r i g i n a l d e t e r m i n a n t could be
r e p r e s e n t e d a s the sum of t h r e e s e p a r a t e d e t e r m i n a n t s (1), (2), and
(3). D e t e r m i n a n t (1) i s s i m p l y the p r o d u c t of F i b o n a c c i n u m b e r s (one
from e a c h row) by the b a s i c power d e t e r m i n a n t .
Thus for the sixth
o r d e r , the situation would be a s follows:
F
BPD
F
l
5
4
5
F's
(1)
F
2
F
3
4
F
F
1
1
1
1
3
2
5
1
1
1
5
4
1
3
1
2
F
6
7
The sign would be n e g a t i v e .
D e t e r m i n a n t (2) can be r e l a t e d to the b a s i c power d e t e r m i n a n t
by moving the f i r s t c o l u m n i n t o the l a s t position.
F o r the sixth o r d e r ,
this involves a change of sign„ Again f a c t o r s can be taken out leaving
a b a s i c power d e t e r m i n a n t .
The p a t t e r n i s a s follows:
F,
F
BPD
r-3
F
5
4
3
F's
1
1
(2)
6
5
The sign would be positive*
2
F
F
2
5
1
1
1
1
1
4
3
2
1
4
6
1964
O F FIBONACCI NUMBERS
89
To evaluate (3) we expand by the f i r s t column.
We shall d e s i g -
nate s u c c e s s i v e e l e m e n t s of the expansion, due account being taken of
a l l signs i n c l u d i n g t h e negative quantities in the f i r s t column, by s u c cessive
capital l e t t e r s : A, B, C, D, . . . . A gives r i s e to a simple
b a s i c power d e t e r m i n a n t ; B to one with a line m i s s i n g
the second line m i s s i n g
(k = 1); C with
(k = 2); e t c . However, t h e r e a r e f a c t o r s that
have to be multiplied in each c a s e . It should be noted too that we a r e
r e f e r r i n g to d e t e r m i n a n t s of the fifth o r d e r and not of the sixth.
F
l
4
3
l
3
l
F
3
2
3
2
i
3
E
2
3
F
D
F
F
F
3
2
3
3
F
2
3
4
F
F
2
2
4
F
F
5
6
F
2
2
2
1
4
3
2
2
1
F
2
3
F
4
F
F
5
6
F
1
1
1
2
2
2
1
3
3
3
2
1
F
7
1
2
2
2
F
3
F
4
F
2
1
1
1
2
3
3
3
F
3
F
4
F
F
5
F
2
1
2
1
F
5
6
6
(negative)
7
(positive)
7
F„
2
2
1
2
1
1
2
1
3
2
4
3
2
2
1
F
3
F
3
F
2
F
1
F
1
F,
F
2
2
4
3
3
3
1
2
1
1
(positive)
(negative)
1
1
(negative)
90
SOME DETERMINANTS INVOLVING POWERS
F
F
l
2
F
•
F
3
F
4
F
5
4
3
2
1
2
2
2
1
4
3
4
3
2
1
F
6
April
7
(positive
S u m m a r i z i n g in one table (omitting the f i r s t and second Fibonacci
number f a c t o r s a s they a r e both unity) we have the following for the
evaluation of the d e t e r m i n a n t of the sixth o r d e r .
Sign
F
(1)
(2)
-
4
+
A
-
B
F
3
F
4
F
5
6
1
F
7
3
2
4
3
2
1
4
3
2
2
1
3
3
3
2
i
C
+
+
3
3
3
2
1
D
-
4
3'
2
2
1
E
-
4
3
2
1
1
F
+
4
3
2
I
The following p a i r s of t e r m s combine:
1
E and (1); F and (2); A and
o
o
D; B and C. The r e s u l t i n g s u m s have a c o m m o n factor of 2 3
2
5 , the adjoint factor being 144. Thus finally the value of the sixth
12 5 2
o r d e r d e t e r m i n a n t is found to be 2
3 5 .
DETERMINANT OF ORDER 12
Without justifying a l l the i n t e r m e d i a t e s t e p s , the s u m m a t i o n
table for o r d e r 12 is shown below*
Sign
F
3
F
1
4
D
F
6
F
7
F
8
F
9
F
10
F
(1)
+
10
9
3
7
6
5
4
3
2
n
£F
F
^13
1
1
12
(2)
.
10
7
4
3
2
1
10
8
7
6
6
5
+
9
9
8
A
5
4
3
2
2
1
B
-
9
9
8
7
6
5
4
3
2
I
C
-
8
8
7
6
5
4
3
2
1
D
•t
9
9
3
4
8
7
7
6
5
5
4
3
2
1
1964
O F FIBONACCI NUMBERS
Sign
F
3
E
+
F
-
G
-
H
I
+
+
9
9
9
9
9
J
-
10
K
-
10
L
.+
10
F
F
4
F?
F
8
7
6
6
S
7
6
8
7
7
8
8
7
9
9
9
9
8
7
8
7
8
7
8
7
5
•
6
6
6
6
6
6
6
6
91
6
b
4
F
12
- 11
2
3
6
5
i
3
5
5
4
3
5
4
5
4
4
5
5
5
F
8
F
9
F
•Mo
13
1
2
2
1
2
1
3
3
3
2
4
3
2
4
"2,
7
2
1
1
1
4
3
2
1
It will be noted that the following p a i r s add up to z e r o :
and G; C and H; B and I; A and J; (1) and K; (2) and L.
value of the determinant is z e r o .
F
1
1
E and F; D
T h e r e f o r e , the
The s a m e r e s u l t w a s found for o r -
d e r s 4, 8, and 16.
DETERMINANT OF ORDER 1!
Sign
(1)
(2).
+
+
F
3
11
11
A
-
11
B
-
10
C
10
D
+
+
10
E
-
10
F
-
10
G
+
+
10
H
I
-
10
J
-
10
K
+
11
'•+
11
L
M
-
10
11
F
4
F
5
10 9
10 9
F
6 F?
F
8
F
9
F
F
10
ll F
8
7
6
5
'4
3
2
8
7
6
5
4
3
2
5
4
3
F
12
13
1
1
F
l
1
•'
2
1
2
1
2
I
10 9
10 9
8
7
6
8
7
6
5
4
3
'2'"
3
9
9
9
9
9
9
9
9
8
7
6
6
4
4
3
8
8
7
6
5
5
4
3' ' \2
1 '
8
7
7
6
4
3
2
1
8
7
6
7
5
4
1
7
7
6
5
2
8
8
7
6' 5
3
2
1
1
9
10 9
10 9
8
7
6
5
5
4
4
4
3
3
2
8
6
6
6
4
3
2
l'
8
7
6
5
4
3
8
7
6
5
4
3
3
2
2
i
2 . 1
10 9
10 9
8
8
7
7
6
6
5
5
4
4
3
3
,2
2
•
•
1
1
1
92
SOME DETERMINANTS INVOLVING POWERS
April
It will be noted that the following p a i r s add up to z e r o : E and G;
C and I; A and K; (2) and M.
The o t h e r s can be combined to give the
following t a b l e .
Sign
F
F
F
9
9
7
6
5
4
3
2
7
6
5
4
3
3
2
1
8
7
6
5
5
4
3
2
1
8
6
7
6
5
4
3
2
1
(1), L
B, J
+
3
12
-
11
10
D, H
+
11
F
-
10
9
9
4
10
10
11
12
1
13
F
14
1
After taking out the c o m m o n factor
10 9 F 8 7 6 6 5 4
*3
4 5 6 7 8 9 10 11 12 13 14
the following r e m a i n s for evaluation:
Sign F 3
F4
F5
F6
F?
+
2
1
1
1
1
-
1
1
1
1
1
+
1
1
1
FQ
1
F()
1
F^
F
n
F
12
F
1
1
1
1
1
1
1
1
1
1
13
F
14
No e a s y method was found for evaluating the sum of t h e s e q u a n t i t i e s .
E s s e n t i a l l y it w a s a m a t t e r of evaluating t h e m ,
then factoring.
combining t h e m and
F o r t u n a t e l y , a s the n u m b e r s to be f a c t o r e d i n c r e a s e d
in size going up to 23 digits in one i n s t a n c e , a p a t t e r n involving F i b onacci and Lucas n u m b e r s w a s d i s c o v e r e d with the r e s u l t that the
f o r m u l a s (1) and (2) on page 38 of [l] w e r e d i s c o v e r e d .
The m a t t e r can be allowed to r e s t h e r e .
The path p u r s u e d h a s
been i l l u s t r a t e d in sufficient detail to allow o t h e r s to explore t h e s e
interesting determinants.
The f o r m u l a s obtained a s well a s the d e t e r -
minant values to the twentieth o r d e r a r e set forth in the p a p e r [l] and
need not be r e p e a t e d .
REFERENCES
1.
B r o t h e r U. Alfred, " P e r i o d i c P r o p e r t i e s of F i b o n a c c i S u m m a 2.
t i o n s , The F i b o n a c c i Q u a r t e r l y , 1(1963), No. 3, pp. 3 3 - 4 2 .
T. L„ B r e n n a n " F i b o n a c c i P o w e r s and P a s c a l ' s T r i a n g l e in a
M a t r i x " The Fibonacci Q u a r t e r l y , 2(1964), No. 2, pp. 9 3 - 1 0 3 .
xxxxxxxxxxxxxxxxxxxx
FIBONACCI POWERS AND PASCAL'S TRIANGLE IN A MATRIX - PART I *
T E R R E N C E A. BRENNAN
Lockheed M i s s i l e s and Space Co., Sunnyvale, California
1.
INTRODUCTION
The m a i n p o i n t of this p a p e r is to display some i n t e r e s t i n g p r o p e r t i e s o f t h e (n+1) X (n+1) m a t r i x P
defined by imbedding P a s c a l ' s
t r i a n g l e in a s q u a r e m a t r i x :
0 0 0 1
0 0
11
0 12
P
(1.1)
The m a t r i x P
n
=
13
1
3 1
was o r i g i n a l l y c o n s t r u c t e d by the author in o r d e r
to evaluate a d e t e r m i n a n t p r e s e n t e d by B r o t h e r U. Alfred.
The d e -
t e r m i n a n t , and its origin, h a s subsequently been published in [l] and,
for the sake of c o m p l e t e n e s s ,
2.
its evaluation will be p r e s e n t e d h e r e .
THE P R O B L E M AND ITS SOLUTION
THE PROBLEM:
Evaluate the fifth o r d e r d e t e r m i n a n t
5
+ 1 - ]1
141
I3 I2
1213
1
5
5
L
2° + 1 - 1
5
5
3 + 2 - ]I
241
2312
2213
2
342
3322
3223
3
- .L
543
5332
5233
5
5
5
8 D + 5 - .L
845
8352
8253
8
1
(2.1)
5
5
5
+ 3
5
" P r e s e n t e d originally at the R e s e a r c h Conference of the Fibonacci
A s s o c i a t i o n , D e c e m b e r 15, 1962.
93
94
and its n - t h o r d e r generalization.
t h e f i r s t c o l u m n w o u l d be
u
April
FIBONACCI POWERS and PASCAL'S
,,
n+1
and
u
n
n
F o r the n~th o r d e r the p o w e r s in
and the
in the l a s t row, w h e r e
d e t e r m i n a n t would extend
u
n
to
is the n - t h F i b o n a c c i n u m -
ber:
u
n+1
(2.2)
+ u
n
with
,
n- i
u;
0
0, u ,
T h e - d e t e r m i n a n t (2. 1), w h i c h w e w i l l c a l l . D_
(D
in generaL),
w i l l be e v a l u a t e d a s a n e x p a n s i o n of c o f a c t o r s of t h e f i r s t c o l u m n . In
o r d e r to k e e p t r a c k of t e r m s i n t h e e x p a n s i o n it i s c o n v e n i e n t t o d e fine D
f o r a n a r b i t r a r y s e q u e n c e a„« a. , ' a~, . . . . a ,'
by a p J
r
n
0
1 2
n+1
p r o p r i a t e l y p l a c i n g t h e m e m b e r s of t h i s s e q u e n c e i n t h e f i r s t c o l u m n
of
D :
D 5 {a}
(2.3)
a
2
a
3
a
4 '
1i
For
a,
i
5
1
i4i
ar
,5
1 5
u a } - 1 aQ
ar ~
C l e a r l y (2. 1) i s
5
5
a
,J
a,
D r {a} w i t h
l
241
L
a^
4
3 2
3
a
4
5 3
5
a,r
845
a n = a, = a~ = . . . = a , = - 1.
s i m p l i c i t y w e w i l l c o n t e n t o u r s e l v e s w i t h t h e r e d u c t i o n of
t h e fifth o r d e r d e t e r m i n a n t
(2. 3) w h i l e m e n t i o n i n g t h e c o r r e s p o n d i n g
r e s u l t s for the g e n e r a l c a s e .
T h e r e d u c t i o n r e s t s on t h e g r o u n d w o r k
of B r o t h e r U* Alfred,,
THE SOLUTION:
.
B a s i c to t h e r e d a c t i o n i s t h e d e t e r m i n a n t of t h e f o l l o w i n g m a t r i x :
r 4
A
2 ~~
(2.4)
Br
\ .J
\ .4
j 8~
i 3 .i
I2!2
1
3
2 1
2212
2
3
? ?
3 2
3
5232
5
3 2
3
5 3
3
8 5
2
8 5
2
8
3
il
1964
TRIANGLE IN A MATRIX
95
and in g e n e r a l B . w h e r e n is the o r d e r of the m a t r i x and i de~
°
ni
notes the f i r s t row e n t r i e s a s u. ,, and u..
i+l
i
An i n t e r e s t i n g p r o p e r t y of the d e t e r m i n a n t of B - . is that its
D j J.
magnitude is independent of the index, or s t a r t i n g point,
i.
This fact
is evident when we m u l t i p l y the two m a t r i c e s
B c . QA - B c . . ,
5S I 4
5$ l+l
(2.5)
where
(2.6)
Q
4
=
1
1
1
1
1
4
3
2
1
0
6
3
1
0
0
4
1
0
0
0
1
0
0
0
0
is the m a t r i x of (1.1) " t r a n s p o s e d " about its counter diagonal.
Since
the d e t e r m i n a n t of Q . is ±1 we have
1
5, i '
5, i+l
M o r e p r e c i s e l y we can develop
l,
(2.7)
IQ,
Q
'n- 1
-i,
lQ 2 h
1,
(-i) n ( n - l ) / 2
We can s t a r t , then, with B r „ and shift indices on each row to obtain
b, (J
(2.8)
But
B
5,0Q4=B5,i
•
96
April
FIBONACCI POWERS and PASCAL'S
,4
B5,0
0
0
0
i3i
I
!
2
231
221
2
^
2
3
5^ 3
2
1 • 1*
2
? 2
5- 1
P a s s i n g to d e t e r m i n a n t s w e h a v e
* 5 f i « -= - Q4, '
where
1
B^ ~
D,
.B
- 5c i 0 l
=
t-l)
1 0 i
l-
2 - 3 - 5
h a s b e e n e x p a n d e d byJ c o f a c t o r s
0 '
common row factors have been removed.
c u r s i v e p r o c e s s for e v a l u a t i n g
|B . |
|B4>i
of i t s f i r s t r o w a n d
Having established a r e -
the g e n e r a l f o r m u l a
maybe
shown:
1B
. = (-1)
v
m'
where
u
, u
u
„ .
n - 1 n - 20
n-3
n-2
s = n(n-l)(3i + n-2)/6 .
In a n o t a t i o n w h i c h w i l l b e m o r e c o n v e n i e n t t o u s e , w e d e f i n e
S
o
= 1,
S ,. = N( - l ) n S
n+1
'
n
F~(x)
=1,
0V '
F
W (x) = F
(F(x))
for
. . (x) = x . . F v(x)
n+1
n+1 n
n > 0
f o r a n yJ s e q u e n c e / x )
^
^ n>
Then
(2.9)
Q .-l
= S
and
n
IB .1 = S 1 - 1 F (S)W , (u)
' ni'
n
n
n-1
Let us s e e w h a t p r o g r e s s c a n be m a d e with
|Dn |
.
W r i t i n g ( 2 . 3) a s
t h r e e s e p a r a t e d e t e r m i n a n t s on t h e f i r s t c o l u m n w e h a v e ,
symbolically,
1964
(2.10)
TRIANGLE IN A MATRIX
97
D 5 {a)
The f i r s t d e t e r m i n a n t is the h a r d one. The second c o m p a r e s nicely
with |Bp. , I after a common factor is r e m o v e d from each row. The
i
third becomes
[Br -. | when the f i r s t column is moved to the last (a
change in sign for an even o r d e r d e t e r m i n a n t ) and c o m m o n f a c t o r s a r e
r e m o v e d from each row.
D.H
Hence
"al
F
6< u > l B 5 , l l - a 0 F 5 < u >
lB5,l
and using (2. 9)
(2.11)
D
.{a) =
-a x F 5 (S) F 6 (u) W 4 (u)
-a Q F 5 (u) W 5 (u)
Expansion of the f i r s t d e t e r m i n a n t by cofactors of its f i r s t column
gives r i s e to d e t e r m i n a n t s of the form
April
FIBONACCI POWERS and PASCAL'S
I3
3
2
( 1 - 2 - 5 - 8 ) ' (I 8 1- 3-5)a.
(2. 12)
5
I2 1
?
Z 1
3
5
^3
the m i n o r being s i m i l a r to
ing.
2
3
82 5
1 • I2
2
2 • \L
I3
3
1*
32
33
8 • 52
53
5
| B . •, j except that the t h i r d row is m i s s -
Our t a s k is to evaluate these m i n o r s .
Starting with
BJ 5. , I
(2.4)we form Ihe p r o d u c t
3
U
n3
0
B
2. 13)
5, 1
Q
4
1
u ,
3
0
-2
2
u ,
2
2
131
U
n2
I
0
~l
0
2
2 1
-l
U
-2
0 • u"
1 • 0"
4~1
u_2l
-1
4
2
1 1
22 I 2
3
U
2 • r
1"
H e r e , as in (2.8), the m a t r i x Q shifts indices on each r o w a n d , o v e r applying this shift,
i n t r o d u c e s the negative side of the Fibonacci s e -
quence by way of the r e l a t i o n u
, = u ,, - u . Expanding the d e t e r m i -
nant of (2. 13) by the t h i r d row we have
(2.14)
|B5>1i
|Q4I"3= ( - u V ^ u ^ H i ^
u2)|B4(_2|
where
u
B~ _ is B „ „ v( i . e . , the fourth o r d e r m a t r i x of (2.4) with
4,-2
4, -2
and u 7 in its f i r s t row) and w h e r e the s u p e r s c r i p t 3 denotes
that the t h i r d row is m i s s i n g .
x
(2.15)
B4,-2
3
-l
u
2
-l
3
n2
u
-2
u
-l
u
2
-2
U
3
~2
0
0 u__,
. . 2
0 u ,
3
u^ 1
3
1 .
2
1 0
1* 1
2
1-0
1 . I2
3
0J
l3
1964
TRIANGLE IN A-MATRIX
99
We t r a n s f o r m (2. 15) to the d e s i r e d m a t r i x of (.2. 12) by
< 2 - l6 >
B
P a s s i n g to d e t e r m i n a n t s ,
1
4,i'
l-2Q3 =
B
ll
and combining (2.16) with. (2.14) we have
luTj^TT^r^? |Q P ' 5, I ' *
Using° u - n = ~ ( - l ) u n for the negative
half of the Fibonacci sequence,
©
•
-i
and evaluating the known d e t e r m i n a n t s ^
W U)
, 3 .
12
^
l
;
V
I B 4! , 1, ' | = (-1)
-S- S.
"5
3 F,(S)
4 ' F2(u)F^(u)
The g e n e r a l c a s e , using this technique, rric,y be formulated a s
(2. 17)
\Bl J = ( - D n r S n + 1 S r F n (S)
W (u)
w
r-1
- ^
r
— —
n+1 -r(u)
Two simplifications to (2. 17) a r e in o r d e r :
(-1)
x
S ,, S = S ,,
n+1 r
n+l-r
and
W (u)
F (u)
vu;
F~~T(u)
F nx+l l - r xTuT "- Wn - l,{u)
F r - 1. (TITF™77™T5T
r-lx
n+l-r
It s e e m s a.rr
p p r o pr r i a t e , since F \(u)/ = u • un - I , un - 2 - . . . u2. u1, is
n
n
a f a c t o r i a l type p r o d u c t for the ssequence / u }* that we define the
s p e c i a l i z e d "binomial coefficient"
100
FIBONACCI POWERS and PASCAL'S
April
m • F M V. W = '.M-'-t:]-' •
n-4x
r
We t h e n h a v e
| B r , | = F (S) W AM) S , ,
f n1] .
1
x
v
n, 1 '
n '
n - l ' n + l - r L^-IJ
T h e r e m a i n i n g d e t e r m i n a n t of (2. 10) m a y n o w b e e x p a n d e d , a n d t h e
general case h a s the form
n+1
F J_1 (u)
F (u)
S (-l) r a r J^LL^LL
U
U
r=2
r
r-1
'
r
|B
1 J
n
l l
~ '
or
F
, ( S ) W %(u)
n-1
n '
n+1
S
r=2
v(-I)
1
'
"
S ,,
[~n M a
n+l-r L r J r
.
T h e f i r s t t w o d e t e r m i n a n t s (2. 11) r o u n d o u t t h e s u m m a t i o n n i c e l y f o r
k=l
a n d 2, s o that w e c a n s t a t e
D v{a} = F . ( S ) W x(u)
n >
n-P
n '
or,
n+1
2
A
r=0
( - l ) r S ,.
n+l-r
summing backwards^
(2.18)
Dn{a} M - i r
1
F n _ l ( S ) Wn(u)
"z
r=0
(-l)rSr f ^
At this point w e c o n s i d e r t h e s u m m a t i o n
/:>
T
1 a
L r J
r
0
1QX
(2. 19)
{a} =
n v/
n+1
2
( - l ) r S ["
l a ,.
_0
r L r J n+l-r
and the a s s o c i a t e d polynomial
(2.20)
*
n ( x )
=
n
z
r=0
(
.
1 }
r
s
[n+ljxa+l-r
1
] an+1.
r
1964
TRIANGLE IN A MATRIX
101
F o r the f i r s t few values of n we have
* 0 (x) = x - l ,
2
0 x (x) = x - x - 1 ,
* 2 (x) = x 3 - 2 x 2 - 2 x + l = ( x + l ) ( x 2 - 3 x + l ) ,
(2. 21)
4
3
2
2
?
0 3 (x) = x -3x -6x +3x+l = (x +x-l)(x - 4 x - l ) ,
« 4 (x) = x 5 - 5 x 4 - 1 5 x 3 + 1 5 x 2 + 5 x - l = (x- 1 )(x 2 +3x+l )(x 2 -7x+l)
The f a c t o r i z a t i o n s suggest the r e l a t i o n
<6n(x) = ( - l ) n _ i (x 2 - v n x + ( - l ) n ) 0 n . 2 ( - x )
(2.22)
where
v
is a Lucas n u m b e r , and v = u ,, + u , . (2. 22) m a y
n
n
n+1
n-1
be p r o v e d by induction, and the complete f a c t o r i z a t i o n of 0
comes
from the identity
v
n
= a
+b
,
where a
- a - 1= b
- b - 1 =0
Thus (2. 22) b e c o m e s
0 n (x) = (-l) 1 1 " 1 ( x - a n ) ( x - b n ) 0 n _ 2 ( " x )
=
<ab)n"1
• ( x - a n ) ( x - b n ) 0 n _ 2 (x/ab) ,
and we can c o n s t r u c t
0 (x) =
n
The evaluation of
(2. 18). and (2. 19),
D n {a}
for
n
n
(x-a b
r =0
) .
a Q = 3L^ = . . . = - 1
b e c o m e s , from
102
F I B O N A C C I P O W E R S and P A S C A L ' S
(2.24)
D (a) = (-1)
n l '
The
evaluation
F AS) W (u) 0 ( 1 )
n-1
n
n
of 0 (1)
requires
separate c a s e s ,
Using (2.23) with
(2.25)
n
?
XI ( l - ( - l ) n " r a
r=0
0(1)=
n
the
April
.
investigation
of four
b = - l/a
)= 0
if and o n l y i f n = 4k
When n = 4k + 2 t h e q u a d r a t i c f a c t o r i z a t i o n (2. 22) b e c o m e s
n/2
0n(x) = (i+x)
n
?
(x + ( - i ) r v 2 r + i)
r=l
and
0n(l) = 2
n/2
n
r=l
(v2r + 2(-l)r) .
U s i n g the w e l l k n o w n r e l a t i o n
(2. 26)
For
0 (1) = 2
n
2
v_ = v ?
n/2
7
II v "
, r
r= 1
when
r
+ 2(-l)
we have
n = 4k + 2 .
n = 4k ± 1 w e h a v e , f r o m (2, 22)
*n(x) =
2
II
r=0
( x Z - ( - l ) r + 1 v 2 r + 1 x - 1) ,
when
n = 4k - 1
( x 2 + ( - l ) r + 1 v 2 r + 1 x - 1) ,
when
n = 4k + 1
n-1
0 (x) =
n
r=0
s o that
2 27
< '
>
n--1
"~2~~
^ >
=
2k-1
n
" ^^Zr-l^Zk-l
n
r=0
r=0
when
n = 4k - 1
V
2r+1
'
1964
TRIANGLE IN A MATRIX
103
n-1
T"
,,
2k
r
0 ( i , , n (-D v 2 r + 1 = s 2 k + 1 n
(2.28)
r=0
when
Combining (2. 25),
v2r+1
r=0
(2* 26),
n = 4k 4- 1 .
(2. 27),
(2. 28) with {2. 23), and using the
sign convention
S
= (-i) 1 ^ 1 1 " 1 )/ 2
and
F (S) = -1
only when
n = 4k + 2 ,
we have
D
4k=°>
2 k
k
D
4k + 1
=
W
^
U
4k+l< >
n
V
n
2r+1 ,
r=0
2k-1
k
D
4k-1
= (
l)
~
W
4k-l
(u)
n
V
n
2r+1 ,
r=0
2k-1
D
4k + 2
= 2 W
u
4k»-2< >
n
n
r=0
V
r+1 f
„ / v
n n-1 n~2
2
W W = u u
u~
... u , u =
n
1 2
3
n-1 n
_
II
,
r=l
n+l-r
u
r
REFERENCES
1.
Brother U. Alfred, Periodic properties of Fibonacci summations, Fibonacci Quarterly, 1(1963), No, 3, pp. 33-42.
Continued next issue.
xxxxxxxxxxxxxxxxxxxx
April
104
FIBONACCI GEOMETRY
H. E. Huntley
Below a r e s o m e additional o b s e r v a t i o n s about H u n t e r ' s
article.
w+z-jL_JL_—E
Y
[1]
.- -- 5
If the r e c t a n g l e A B C D h a s a t r i a n g l e /
D P P ' i n s c r i b e d within it so that
AAPD
= V|
ABPP' = AP'DC then x(w+z) = wy = z(x+y)
v^hence
y __ w+z _ z
x
w
w-z
(i)
(ii)
2 2
.
.*.w -z = wz, l. e. ,
F r o m (i)
(iii)
2
2
w -wz-z = 0
+ V7+4z
^>z
~ = *P or y = ^?x
Thus P , P ' divide t h e i r sides in the Golden Section.
Now, suppose ABCD is the Golden Rectangle,
beloved of the
G r e e k a r c h i t e c t s , i.e., A B / B C = <P, then ^ - ^ = <p. Hence, from (ii)
,
x(l + <p)
i. e. x= <pz whence x= w. F r o m (i) y = w+z = (p z.
and(lll)
::=
TfT+?]
*
Since < A = < B r t < and x = w, y = w+z, t r i a n g l e s PAD, P ' B P a r e
congruent. It follows that P D = P P 1 , t h a t < A P D is the c o m p l e m e n t
of < B P ? ' , whence < D P P ' is a right angle.
The a r e a of the right t r i a n g l e is
J(w2+y2)
=
J ( ^ 2 z 2 + / z 2 ) = i / z 2 ( ^ 2 + l ) = iz2(^+l)(^+2)=iz2(4^+3)
we m a y conclude, t h e r e f o r e , that if the r e c t a n g l e is the Golden R e c tangle, that i s , if its adjacent sides a r e in the Golden Ratio, <p, then
the i n s c r i b e d t r i a n g l e is r i g h t - a n g l e d and i s o s c e l e s , the length of the
equal sides being z "w4^+3 .
E d i t o r i a l Note: P P 1 1(1 AC
1.
REFERENCES
Jo A. H. Hunter, " T r i a n g l e I n s c r i b e d in a R e c t a n g l e " 1(1963) Oct o b e r , pg. 66,
ON SUMMATION FORMULAS FOR FIBONACCI AND LUCAS NUMBERS
DAVID Z E I T L I N , Honeywell,
Minneapolis, Minnesota
n
X F , , , where
, ,
ak-b
k=l
a > b a r e positive i n t e g e r s and F, a r e Fibonacci n u m b e r s with
F nu = 0, F ,1 = 1, and F n+2
. 0 = Fn+1
In this note,
(1 + F , n = 0, 1, . . . .
n
we will e s t a b l i s h a m o r e g e n e r a l s u m m a t i o n formula which yields the
Recently, Siler [ l ] gave a closed form for
'
r e s u l t of [ l ] a s a s p e c i a l c a s e .
G e n e r a l s u m m a t i o n f o r m u l a s for F i b -
onacci and Lucas n u m b e r s will be obtained as s p e c i a l c a s e s of our
general result.
Theorem.
Let p, q, u * and u,
be a r b i t r a r y r e a l n u m b e r s ,
and Let
(i)
n+2
n+1
pun
(n = 0, 1, . . . ) ,
n . n
S = r + r
(n= 0 , 1 , . . . ) ,
1
2
n
2
2
w h e r e r , / r~ a r e r o o t s of x - qx + p = 0 (i. e. , q - 4p •/ 0). We define
(2)
(3)
u_n = (uQSn - u n ) / p n
(n= 1 , 2 . . . . . ) ,
(4)
S
(n= 1,2, . . . ) .
-n=Sn/pn
Let a = 0, 1, . . .• d = 0, ±1, ±2, . . . , and let x be a r e a l n u m b e r .
71 „
(1-S
x
a
, a 2.
x + rp x )
v
2
in
k=
(5)
- x
u , ,,x
ak+d
k
Then
a n+2
= rp x
u . j
an+d
°
u•an+a+d
, , -j + xu a+d
, j + x(1-xSa ) ud,
.
M o r e o v e r , in the r e g i o n of c o n v e r g e n c e , we have
a 2 ' *°
k
(1-Sax+P x ) S uak+dx =ud + (ua+d- u d S a ) x
(6)
.
k=0
Proof. If C , i = 1,2, a r e a r b i t r a r yJ c o n s t a n t s , then u = C, r 1 + C 0 r ,
—
l
n
1 1
2 Z
n = 0, 1, . . . , is the g e n e r a l solution of (1). Then
v
k = u ak+d
=
(Clrl)(rl)k
+
<C2r2)(r2)k '
s a t i s f i e s the l i n e a r difference equation
105
k = 0, 1, . . . ,
106
ON S U M M A T I O N F O R M U L A S F O R F I B O N A C C I
<7>
V
s i n c e (x
k+2 =
S
aVk+l "P^k
(*=0,1....)
- S o x + p ) = (x - ^ ( x - r ^ ) .
a
k
S
V x
. k=0
M u l t i p l y i n g b o t h s i d e s of (7) by x
k+2
and then s u m m i n g both s i d e s with
k3 w e o b t a i n
T .0
n
kfZ
_ .
v
= x &
k,Z0 Vk+2X
a = V k + l
•
.
-
r e s p e c t to
n
<»>
.
Let
n
g(x) =
April
, .,
k+1
X
0
a 2
- P
X
n
v
,*
k=0
V
,.
. k
kX
'
k=0
We n o t e t h a t
x
,~,
(9)
k+2
^
vk+2x
k= 0
(10)
=
k=0
V
k+lX
If w e s u b s t i t u t e
, , .
=gWtv
=
^
x ) + V
6 + 2
n+2 .
n+1
x.
-v
+ v n + 1 x
n+l
X
-
V
(9) a n d (10) i n t o (8), u s e (7) to e l i m i n a t e
The g e n e r a t i n g function for
R>0
for
|x|<R^ v x
Remarks.
-v
and suppose that
—>0 a s
s
k=0
v,
V
,
v
?s
(5).
c o n v e r g e s for
k
n->c&<.Thus, for
|x|<R.
Then, f o r u n = 0 a n d u , = 1, w e
~ F , F
= (-1)
F , and S = L , the well-known
n
n
-n
n
n
n
s e q u e n c e , w h e r e L n = 2 a n d L, = 1.
T h u s , (5) a n d (6), f o r u
become, respectively,
(x 1 - L x + ( - l ) a x 2 ) £ F . , , x k = ( - i ) a x n + 2 F
,,
a
i n
ak+d
an+d
k=0
-x
(12)
n+1
F
, , , + x F , , + ( 1 - x L ) F , .,
an+a+d
a+d
a d
(1-Lax + (-l)ax2)
Tiien,
| x | < R , (5) y i e l d s (6) a s n ^ o o .
have u
(11)
and
i s r e a d i l y o b t a i n e d f r o m (5).
L- X
L e t q = 1 a n d p = - 1 in (1).
Q
0 *
s o l v e f o r g(x), w e o b t a i n o u r p r i n c i p a l r e s u l t ,
Let
l X
~
Fak+dxk = Fd + (Fa+d - FdLa)x
k=0
Lucas
= F ,
1964
AND LUCAS NUMBERS
107
The m a i n r e s u l t of [1] is obtained from (11) for x = 1 and d = - b .
F o r x = - 1 , (11) yields the i n t e r e s t i n g r e s u l t
(13)
K-U
+
<-1)nFan+a4d-Fa+d
+
<La+1)Fd "
F o r d = 0, (12) yields
(l-Lax + (-l)ax2)
(14)
T Fakxk= Fax.,k=Q
Again, let q = 1 and p = ~1 in (1).
( a = 0 , 1 / . ...) .
Then, for u n = 2, and u = 1,
we now have u = L , S = L , and L = (-1) L . Thus, with u • = L ,
n
n n
n
-n
n
n
n
(5) and (6) b e c o m e , r e s p e c t i v e l y ,
(l-Lax+(-l)ax2)
(15)
(16)
x
~ '
z Lak+dxk= ("Daxn+2L
k=0
- ^n+1Lan+a+d
4- x L a+d
_ + (x l - x L a' ) dW
( 1 - L x + x( - l ); a x 2 ) IT L , , , x k = L , + x(L , , - L L , ) x .
*
a
i r* • ak+d
d
a+d
a d'
k=0
REFERENCES
1.
Ken Siler, F i b o n a c c i s u m m a t i o n s , Fibonacci Q u a r t e r l y , Vol. 1,
No. 3, October 1963, pp. .67-69..
xxxxxxxxxxxxxxxxxxxx
108
LETTER TO THE EDITOR
The Editor,
Fibonacci Quarterly.
Dear Dr0 Hoggatt,
I refer to the article, "Dying Rabbit Problem Revived" in the December
1963 issue. The solution given there is patently wrong — if only because the alleged number of rabbits tends to minus infinity as n tend
to infinity. It may easily be shown that the correct answer, X , is
given by the recurrence relation
X n o = X ., ~ T X _ . - X
n+13
n+12
ntl 1
n
3
n
0
together with the initial conditions
X = F , . for n = 0, 1, . . . , 11;
n
n+1
X, 0 = 232 .
12
2
13
12
In view of the fact that the two equations z - z - 1 = 0 and z
- z
- z•*••*• + 1 = 0 have no common root, it is clear that the answer can
never be expressed simply as a-linear expression in Fibonacci and
Lucas numbers whose coefficients are merely polynomials i n n . For,
any such expression, Y, where the highest power of n which occurs is
n m , satisfies
(E 2 - E - l ) m + 1 Y = 0 .
In particular the expression found by Bro. Alfred satisfies
(E 2 - E - I ) 2 Y = 0 .
The error made by Bro. Alfred stems from his table on p. 54 where
the number of dying rabbits in the (n+13)th month is seen to be F for
n = 1, 2, . . . 11 and it is then assumed without proof that this is true for
other values of n. In fact the very next but one value on n, namely
n = 13 shows that this is false. In fact of course the number of dying
rabbits in the (n+13)th month equals the number of bred rabbits in the
(n+l)th month, and this will be less than F for all n exceeding 12.
Yours sincerely,
(John H. E. Cohn)
BEDFORD COLLEGE
(University of London)
SQUARE FIBONACCI NUMBERS, ETC.
JOHNH.E.COHN
Bedford College, University of London, London, N . W . I .
INTRODUCTION
An old conjecture about Fibonacci n u m b e r s is that 0, 1 and 144
a r e the only p e r f e c t s q u a r e s .
Recently t h e r e a p p e a r e d a r e p o r t that
computation had r e v e a l e d that among the f i r s t million n u m b e r s in the
sequence t h e r e a r e no further s q u a r e s [ 1 ] .
This is not s u r p r i s i n g ,
a s I have m a n a g e d to p r o v e the t r u t h of the conjecture, and this s h o r t
note is w r i t t e n by invitation of the e d i t o r s to r e p o r t m y proof.
o r i g i n a l proof will a p p e a r s h o r t l y in [ Z ] and the r e a d e r is
The
referred
t h e r e for d e t a i l s . However, the proof given t h e r e is fairly long, and
although the s a m e method gives s i m i l a r r e s u l t s for the Lucas n u m b e r s , I have r e c e n t l y d i s c o v e r e d a r a t h e r n e a t e r method, which s t a r t s
with the Lucas n u m b e r s ,
p e a r s below.
and it is of this method that an account a p -
It is hoped that the full proof t o g e t h e r with its c o n s e -
q u e n c e s for Diophantine equations will a p p e a r l a t e r this y e a r .
I might
add that the s a m e method s e e m s to w o r k for m o r e g e n e r a l s e q u e n c e s
2
4
of i n t e g e r s , thus enabling equations like y = Dx + 1 to be c o m pletely solved at l east for c e r t a i n values of D.
Of c o u r s e the F i b -
onacci c a s e is s i m p l y D = 5.
PRELIMINARIES
In the f i r s t p l a c e ,
onacci Q u a r t e r l y ,
n-th.
in a c c o r d a n c e with the p r a c t i c e of the F i b -
I h e r e u s e the s y m b o l s
and
L
to denote the
F i b o n a c c i and Lucas n u m b e r r e s p e c t i v e l y ; in o t h s r p a p e r s I
u s e the m o r e widely accepted,
[3].
F
Throughout the following
if l e s s logical, notation u
n, m, k will denote i n t e g e r s ,
n e c e s s a r i l y positive, and r will denote a n o n - n e g a t i v e i n t e g e r .
w h e r e v e r it o c c u r s ,
not
Also,
k will denote an even i n t e g e r , not divisible by 3.
We shall then r e q u i r e
the following
formulae,
elementary
(1)
x
'
and v
2F
, = F L + F L
m+n
m—n - n ~ - m
109
all of which a r e
HO
SQUARE FIBONACCI NUMBERS, E T C .
(2)
2 L , = 5 F F + L L
m+n
m n
m n
(3)
L0 ' = L2 + (-I)™"1 2
2m
m
v
(4)
(F-
, L.
) = 2
(5)
( F , L ) = 1 if
(6)
2 L
1
(7)
3>n
if a n d o n l y if
. J
m
3 L
'
J
F_n=(~-Dn-1Fn
(9)
v
'
L
v
< 12 >
F
(13)
L
,
= - L
m + 20k1 " " . . m
a+2k= m +
t
n
Lk=3.(mod4)
L
\
= (-l)nL
-n
(10)
3 m
'
if a n d o n l y if m = 2 ( m o d 4)
m
(8)
(11)
V. '
April
1 2 ^
L
F
if
2'|k,
3 ^
( m o d L. )
.
k
m
^ °
d
L
k>
^m°d8)
m
THE MAIN T H E O R E M S
T h e o r e m 1.
If
L
n
= x
*
then
n = 1 or 3.
Proof.
If n
i s e v e n , (3) g i v e s
= y2 ± 2 / x2
L
n
7
.
If 115 1 ( m o d 4 ) , t h e n L , =• 1, w h e r e a s if n & 1 w e c a n w r i t e n = 1 +
r
2* 3 # k w h e r e k h a s t h e r e q u i r e d p r o p e r t i e s , a n d t h e n o b t a i n b y (11)
L
and so
2
Ln^x^
since
=. - Li = - 1 ( m o d L, )
- l i s
a n o n - r e s i d u e of
L^. by ( 1 0 ) .
Finally,
1964
SQUARE FIBONACCI NUMBERS, E T C .
Ill
-?
if 113 3 (mod 4) then n = 3 gives L, = 2 , w h e r e a s if n / 3, we
w r i t e a s before n = 3 + 2*3 -k and obtain
L n =. -o L
= - 4 (mod Lj-^)
and again
&
L / x .
n
This concludes the proof of T h e o r e m 3.
T h e o r e m 2.
2
If L = 2x , then n = 0 or ±6.
n
Proof.
If n is odd and L is even, then by (6) n = ± 3 (mod 12) and
so, using (13) and (9),.
L
n
= 4 (mod 8)
and so
L ^ 2x .
n
Secondly, if n = 0 (mod 4), then n = 0 gives
r
if n ^ O , n = 2 * 3 »k and so
2L
n
= - 2L
0
2L / y
n
2
•, i . e .
L ^ 2x
n
2
L, -• 2 • 3
Thirdly, if n = 6 (mod 8) then n = 6 gives
if n / 6 ,
whereas
n = 6 4- 2*3 ' k w h e r e now 41 k, 3/fk and so
2L
and again,
a s before
= 2, w h e r e a s
= ~ 4 (mod L, )
k'
2
whence
L
n
= - 2Lx = - 36 (mod L, )
o
k
- 36 is a n o n - r e s i d u e of
L
n
Finally,
L,
using (7) and (10).
Thus
/
^ 2x".
if n =. 2
(mod 8), then by (9)
L
= L
w h e r e now
- n E 6 (mod 8) and so the only a d m i s s i b l e value is -n = 6, i. e,
This concludes the proof of T h e o r e m 2,
11= - 6 .
112
SQUARE FIBONACCI NUMBERS, ETC.
April
Theorem 3.
If
= x 2 , then n = 0,\±1, 2 or 12.
F
Proof.
If n = 1 (mod 4), then n. = 1 gives
r
n = 1 + 2e 3 • k and so
F
whence
n
F, = 1, whereas if n / 1,
= - F, = - 1 (mod L, )
1
k'
F
/• x . If n = 3 (mod 4), then by (8) F
= F
and -n = 1
J
n
"
-n
n
(mod 4) and as before we get only n = - 1 . If n is even, then by (1)
2
F = F
L,
and so, using (4) and (5) we obtain, if F =• x
n
n
i/2n i/2n
2
2
either
3 In, F , = 2y , L
= 2z .
By Theorem 2, the laty
y
'
y2n
i/2n
ter is possible only for i/2n ='0, 6 or - 6 . The first two values also
satisfy the former, while the last must be rejected since it does not.
or
3iri, F , = y , L
= z .
By Theorem 1, the latter
'
Vi*
V2n
is possible only for i/2n = 1 or 3, and again the second value must
be rejected.
This concludes the proof of Theorem 3.
Theorem 4.
If
F
= 2x Z , then n = 0, ±3 or 6.
n
Proof.
If n = 3 (mod 4), then n = 3 gives
F~ = 2, whereas if n ^ 3,
n = 3 + 2" 3 * k and so
2F
and so F
get
B
only
y
n
n
= - 2 F a = -4
3
(mod L, )
k
4 2x . If n = 1 (mod 4) then as before
n = -3.
must have if
F.
If
z
n
is
even,
then
since
F
= F
and we
-n
n
F = F,
L
we
n
i/2n y2n
= 2x
2
= y ,
/2n
2
F
L,
= 2z ; then by Theorems 2 and 3 we
i/2n
i/2n
see that the only value which satisfies both of these is y2n - 0
2
2
or
F
= 2y ,
L
= z ; then by Theorem 1, the second
J
3
i/2n
i/2n
'
either
of these is satisfied only for' i/2n = 1 or 3.
does not satisfy the first equation.
But the former of these
1964
SQUARE FIBONACCI NUMBERS, E T C .
1.13
This concludes the proof of the t h e o r e m .
REFERENCES
1.
M. Wunderlich,
On the n o n - e x i s t e n c e of Fibonacci S q u a r e s ,
Maths, of Computation, 1_7 (1963) p. 455.
2.
J. H. E, Cohn,
On Square Fibonacci N u m b e r s ,
Proc.
Lond.
M a t h s . Soc. 3^9 (1964) to a p p e a r .
3.
G. H. H a r d y and E. M. Wright, Introduction to Theory of Numb e r s , 3rd. Edition, O. U. P . 1954, p. 148 et seq.
XXXXXXXXXXXK< KXXXXXX
I
/
EDITORIAL NOTE
B r o t h e r U. Alfred cheerfully acknowledges the p r i o r i t y of the
e s s e n t i a l method, u s e d in " L u c a s S q u a r e s " in the last i s s u e of the
Fibonacci Quarterly Journal,
r e s t solely with J. H. E. Cohn.
This
was w r i t t e n at the r e q u e s t of the Editor and the unintentional o m i s s i o n
of due c r e d i t r e s t solely with the E d i t o r .
114
April
EXPLORING THE FIBONACCI REPRESENTATION OF INTEGERS
P r o p o s e d by B r o t h e r U. Alfred on page 72, Dec. 1963,
!!
The Fibonac ci Q u a r t e r ly ! f
The completion of the T h e o r e m stated in the a r t i c l e i s :
The Maximum n u m b e r of different Fibonacci ' n u m b e r s r e q u i r e d to
r e p r e s e n t an i n t e g e r N for which [iMj * = P^ is given by -^ L
This is a c o r o l l a r y of the following t h e o r e m .
For
F
< N £. F , , the n u m b e r N can be r e p r e s e n t e d a s a' sum of
n
n+1
F i b o n a c c i m i n i b e r s s the 1 a r g e s t which is F and the s m a l l e s t g r e a t e r
than or equal to F 9 . M o r e o v e r , the sum n e v e r contains two c o n s e c u tive Fibonacci n u m b e r s , 'We t h e r e f o r e have at m o s t the a l t e r n a t i n g
as c l a i m e d .
t e r m s of indices from 2 to n which gives us
The proof of this t h e o r e m depends upon a L e m m a which is a well
¥+']=[!]•
known Fibonacci Identity that
F~ + F , + F , + . , . + F~ = F~ , 1 - 1
Z
4
6
Zn
2n+l
and that P 0 + F r + F„ + . . . + 'B\ , = F n - 1. The proof of the f i r s t
3
D
{
In-1
Zn
p a r t of this is given by induction and the second p a r t is s i m i l a r l y p r o v e d .
Proof,
J
F o r n = 1, we have P 7 = ¥\ - 1
n = 2, we have F ? + F . = F r - 1 which c l e a r l y shows the L e m m a
holds for 11 = 1, 2ff,
Now a s s u m e that it holds for a i l n <. K? w h e r e K is a fixed but u n specified positive i n t e g e r g r e a t e r than or equal to 3.
i . e . P.^ + F , + . . . + F \ _ = F » T , . . - 1, t h e r e f o r e by addition to both '
2
4
ZK
2K+1
sides we have that F n + F" + . . . + F-,,- + F,,,T7.,.., = F~ _ _,v + F O T f ..^ - T"
2
4
2K
ZK+2
2K+1
ZK+Z
= F
- 1
M
2K+3
which i m p l i e s the L e m m a holds for a l l positive n,
Using this Lemm-* which we shall c.?U L e m m a 1, p a H A ion the
f i r s t c a r t whica was last proved, arid p a r t B for the second p a r t with
the odd \ n d i c e s ; we c-»,n now prove the g e n e r a l t h e o r e m that for
F < N . I F ., , we car. r e p r e s e n t N es a s\im o£ at l e a s t a l t e r n a t i n g
rj
n M"
'
.
Fibonacci nurabe L'L- w h e r e the l a r g e s t is F for N < F , - and which
°
a
n+I
t r i v i a i l v is iu;'t F . , iC^eLf when IN - F . .
n+J
nid
Proof. F o r N - i ,' we have 1 - f\«2' aiulior N = 2» we have 2 = F o3s Now
a s s u m e the t h e o r e m t r u e for a l l M <. k» where- k i H a fixed but unspecified
positive i n t e g e r and n is such that F < k 5. P . , , n > 3*
°
n
n+1
—
Continued on Page 134
r
Now if
PARTITION ^NUMLI'AM'^ l\
v irA^$
D'^Uk
i
GeoiMG in i .<* il ,
i ILLL CR
r u l t v " , #,< , n t a / G c .
Netto [1] illustrate 3 a ,,i(~n> ,
n having exactly
\Jf AMPLER PART?TJOMS
1
<"c 5- ?nur aerating ai* partitions of
!
It i-? shown he re m that
f
. n cJgoritliaur form through ase
p m o n i l o i s , .» ' «oi»- . t i c
Netto's procedure can be r e i n ** *
of simpler partitions wnlzh ? ^ h' • * .4 IJJ range (size and number of
members) but otherwise uru _ > ^-!'-d5
limited partitions per sc an ' a *
procedure
of these range-
Mu? of adapting them to Netto's
enumeration proceoura ore din en >J
algorithmic
Properties
w
I
13 Mv-, d >^»
An "Uviouc application of the
;ofn j'ltd tlon
of both lyp^s of
partitions.
2.
LIMITED-RAi ICI", JN * \,t TiJiC I~T?D PAR I IT JONS
Chr y s ia, 1! s [ ?, ] part it ic " \ e 111 l^wf,/, suit? blymodifled. is u r e d
th r oughout.
To u &,
P t -£ J1.
-- "> -, ,
rO
e n u m e r a t i o n o 1 p a r t i ! io "> ^ o T ^ . u
elusive, no partition ba " £ j e. t
each member being not >^-i
I
- c -tiV' i'lL-^ers from
spe» m e s tJtie
11 tc n v in-
i»„ uoi ir^ore than M. m e m b e r s ,
.'an
,
1 .H m ire than q ? .
rio^' = v e i ,
the set rather ib^ii the CPUUJ >du^ ^ :f i ••nge-'imited partition^ is of
immediate inte r e st ir .'hI s pa • >, ^ a* c fo .^rcifyi sff a
x
- •;» :^-> " 3 - / ,
to the enumeratic»n rotation
V is appended
P ^ i ^ n ^ <r.n^ | ^ p l f 4 p J
=^qi , 4q ? ) denotes the 5 ^t oi ,< -i ior. " ' u v ^ g rbe properties of the
enumeration counterpar c,
The existence cu^/'jLioor t« ^
q p , 4 n ^ simultaneous I -y.
Mc; *
are optimum extreme / ? l , e
c
'
£
Brackets [ J except W J ^ I P o r r c
1
customary manner "VVIJILX^ ^ 1 in 1
less
4
than or equal ro in*
Heaslet [3] .
1 * rt )jt.
V=pr
f[
2p2^nr
j "V "ixed ?, , n 0 , q,, q^s there
am
i l iese a re
u< r rcfeicncf s a i e r^c-d in the
IJ i'id„( a i e itie
t* <-} ^ eted.
0 ? t-ai«
^ integer
See U^p^nsky and
116
PARTITION ENUMERATION BY MEANS
(1)
P l opt_
W
If
p,
P2 opt.
ApriL
= " [-(n/q^l .
= [
Vql
]
•
p
l - p l opt.* p l C a n b e c h a n 8 e d t o Pi 0 p t . ' b u t i f P i > P ! o p t . '
cannot be changed. However, if p 7 —P?
can be changed
f > P?
to p ?
, but if p
Lt O p t .
p7
, p„
In generating the partitions,
found first, then the
cannot be changed.
&
tit O p t .
Ct
the
p, -member
partitions are
(p, +l)-member partitions, e t c . , until the p ? -
member partitions are found.
The procedure used herein for the par-
titions of a typical p-member
set is as follows:
A trial "first" partition is formed from
the p m e m b e r s i s equal to or greater than n
to n ? , the partition initiates the set.
p q, ' s .
If the sum of
but less than or equal
If such is not so, the right-
hand member is augmented so that the sum of the p-members
To form new partitions,
creased by one until either it equals
equals
n?
(or both).
in n, .
the right-hand member is successively inq?
or the sum of the p
The next p-member
members
trial partition is found by
adding one to the member second from the right and replacing all
members to the rightwith the new value of the changed member.
The
desired reinitiating partition is found from the sum of the p members, as before.
The right-hand member is successively increased
by one to form new partitions.
When the possibilities of the particular
second member from the right are exhausted,
one is added to the
third member from the right and the process repeated all over again.
Eventually, all p-mernber
ample for
partitions will be accounted for.
PV(>8, <10 |>2, <5 |>2, <7) follows:
2.6
2.7
3,5
3, 6
3,7
4.4
4.5
4, 6
5,5
2,2,4
2,2,5
2,2,6
2, 3, 3
2,3,4
2,3,5
2,4,4
3, 3, 3
3,3,4
2,2,2,2
2,2,2,3
2,2,2,4
2, 2, 3, 3
2,2,2,2,2
An ex-
1964
OF SIMPLER PARTITIONS
3.
117
APPLICATION OF NETTO'S METHOD
Netto [ 1] considers the enumeration
P(n|p|<q)
of the parti-
tions of n having exactly p members with no member greater than
q.
Netto's methodis limited to q>(n+l-p)
tions being p<n and qp>n,
with the existence condi-
simultaneously.
In the terminology of
this paper,
(3)
where
P(n|p|<q)
-
ta = 0 , 1 ,
S [ i ( n - p + 2 - 3 t r 4 t 2 - . . . -pt
~ j*y
.
2)]
,
Inspection of (3) reveals that the
typical term is
n-p+2-w
(4)
in which w is always zero for
ta - 0, always 3 for
ways greater than 3 for all other
t ' s.
t
= 1, and al-
It can be observed that ex-
cept for the zero value of w, each w in the enumeration
P(n|p|<q)
is the sum of the members of each partition included in the set
P V ( > 3 , < n - p | > l , < [ ^ E ] |>3, <p)
(5)
Thus, except for
(6)
p = 1,
P(n|p|<q) =
n-p+2
2
n-p+2-w.
+ 2
i
It should be noted that (5) does not exist for
There are no w.'s
l
(n-p)<3.
under these conditions, and the summation term
of (6) is accordingly zero.
(7)
p = 2, and/or
The special case of p = 1 is
P(n 11 I <q) = 1
118
P A R T I T I O N E N U M E R A T I O N BY M E A N S
Vs w a s s t a t e d f a i b ^ ,
(n
ru^lu^
risscxihtd
t i c u l a x l y a d a p t a o i e tu > g i f a i . ' rnp . r j ' a , s .
April
herein are par-
To ilias end ? t h e a u t h o r
cars s u p p l y -d i i m i t e a u r a b c r *». " I ^ O S U n g u a g e p r o g r a m s a n d t e s t
e x e m p l a r s for e n u m e i d L n ^ h i h l JS
with t h e B u r r o u g h s ZZO d i g i t a l
computer*
ft1] P E R B" 11C D 3
1.
E„ N e t i o , ^£tLfJ£UC h - J P I C J T?J>IJL=I l<' ~K 5 L e i p s i z ,
2..
G, C h r y s t a l ,
T ^ U J O O ! ^ 2LJ±*JeL!S"
Publishing Co.,
3
V
"°'*
2j
1901, pp. 127,
(Reprint)
Mo^ v i K, i ;b^
J . V. U s p e n s k y -oi 1 M A . h ^ r s i 1 * M c G r a w - H i l l B c o k G~, , >-*
I ler^ntaxy
Number
x 1% 1 9 3 9 , p p . 9 4 - 9 9 .
xxxxxxxxxxxxxxxxxxx*
i ^ M I ^ J I P L V : „ 1 1P-JLI^LLM gJ.A._TJ-Q._3
P a g e 44;
On l i n e 4 r e s d "0 ^ k ;„ ^
P a g e 4-9;
On l i n e 8 :>:—id \i\ir
P a g e 80:
In B - 7 l i n e 2 x = 1/4
VV
and
I
J or
!
£<»r "0 £ k < 2 r "
I r»F 1 F
s Tri2 / 4 i
L2
i ^ O ^ i ^ ~ 25
?
'
£ I I 5 ^ ^ i i ^ i l : : M i ^ r L f PV JLf ':_./OPPME1, NO. 4
Reference 4
In H - 2 5
Chelsea
(
H i e i i r i a u t h o r :,- i^v^L^cLVJJN ,
(i, j = 1, 2, 3 , 4 )
Theory,
A NOTE ON WARING7S FORMULA FOR SUMS OF LIKE POWERS OF ROOTS
S.L. BASIN
Sylvania H e c t r o n i c Systems, Mountain View, California
k
k
k
S, = x " + x'~ + . . . Jr x' u m a yJ be e xlp r e s s e d in
k
I
2
n
t e r m s of e l e m e n t a r y s y m m e t r i c functions or in t e r m s of the coefficients
Sums of rp o w e r s
of:
f(x) = (x-x, )(x--x 9 ) . . . (x-x ) = x'"+p,x ' " + . . „ +p^
J.JL
i^i
A,
X
XX
by Newton 1 s f o r m u l a s , u s u a l l y introduced in. a c o u r s e in the t h e o r y of
equations, for e x a m p l e , J . V. Uspensky ll] .
The r e l a t i o n s h i p between Waring\s formula for s u m s of like
p o w e r s of the r o o t s of a q u a d r a t i c and Lucas n u m b e r s is quite obvious
although p e r h a p s a little too s p e c i a l i z e d for L, E. Dickson. \Z] to have
pointed this out in his text, F i r s t C o u r s e in the Theory of Equations.
In o r d e r to obtain an explicit e x p r e s s i o n for
S,
where
k = 1,
k.
2, 3. . . , f i r s t c o n s i d e r the q u a d r a t i c
2
x"
(1)
T
If we denote the r o o t s by a
(2)
px
T
q = 0 .
and & then (1) m a y be r e w r i t t e n a s
x " + px + q = (x~a)(x--/3) ,
After m a k i n g the t r a n s f o r m a t i o n
x = 1/y
and multiplying by y "' we
obtain^
(3)
I + py + qy" = (1-ay){l-/3y)
.
Differentiating both sides of (3) vvHii i e s p e c t to y and dividing
b o t h m e m b e r s of the differentiated equation by the c o r r e s p o n d i n g m e m b e r s of (3) we a r r i v e at
119
120
A N O T E ON WARING'S F O R M U L A F O R SUMS
April
Equations,
k+1
_,_ 2 ,
, k k - 1 , <*
k
-=a+o: y +. . . +a y
+ - —y
J
1-ay
^
1-ay 7
a
(5)
N
'
and
k+1
a r e both obtained f r o m the g e o m e t r i c
(7)
v
'
=
T^—
1-r
k-1
2
. n
series
rJ + 1
1-r
J=0
for e x a m p l e ,
l e t r = a y a n d m u l t i p l y b y a . A d d i t i o n of (5) a n d (6)
r e s u l t s in.,
)
(8
v
'
_«_+T4_=S1+S?y
1 - a y 1-py
1
2'
k+1
+
where
k+1
. . . +S7v y k - 1 + «—i^M+^Iil^l)
k
S, = a
k
+ /?
(1 ~ a y ) ( l - / ? y )
k
In o r d e r t o e x p a n d t h e l e f t - h a n d m e m b e r of (4) u s i n g (7), l e t r = - p y
~qy , t h e n
k_1
(9)
k k
l
—z = & ( - D W q y 2 ) j + ( - p - q y ) I
1+py+qy
1+
Py-qy
E m p l o y i n g the b i n o m i a l t h e o r e m we m a y w r i t e
( P y+qy 2 ) j = 2 ( - f ^ (py) g (qy 2 ) h
1964
O F LIKE POWERS OF ROOTS
121
w h e r e the s u m m a t i o n i s taken o v e r a l l t w o - p a r t p a r t i t i o n s of j , i . e . ,
for a l l g>0 and h > 0
-P" 2 c iy
1+py-qy
(10)
such that g + h = j .
, ^
,
v /
l x g+h+l
Therefore,
(g+h).r
*•
g h g+2h ,
k k
(-p-2qy)(-p-qy) y b ^
1+py+qy
Now the left-hand m e m b e r s of (8) and (10) a r e equal a s shown
by equation (4); t h e r e f o r e we m a y equate coefficients of like p o w e r s of
k-1
y. Specifically, equating coefficients of y
we a r r i v e at
(11)
w h e r e w e have r e p l a c e d
i for g + 1 and j for h in (10). The s u m -
m a t i o n s in (11) now extend over all i>0 and j >0 such that i + 2j = k,
Combining both s u m m a t i o n s in (11) we have,
(12)
sk=kZ(-l)i+J(i^PV
s u m m e d over all i > 0 , j^O such that i + 2j = k.
-1 we have S, = L, , the kth
(13)
L
k
-k
C l e a r l y , for
p - q
Lucas n u m b e r ; and (12) b e c o m e s
T
^
i>0
j>0
i+2j=k
( i + J " 1 ) i~iTjT
122
A NOTE .ON.WAKING'S FORMULA FOR SUMS
April
Equation .(12) of Waring 1 s•• formula, published in 1762, which can
be extended to. include the sum of kth p o w e r s of the r o o t s of
nth
degree polynomials.
The m a i n point to o b s e r v e is not n e c e s s a r i l y the m e a g e r r e s u l t
given by (13) but the fact that i m p l i c i t in the development of W a r i n g ! s
formula lies the generating function (4) for the Lucas n u m b e r s .
REFERENCES
T h e o r y of Equations,
1.
J. V. Uspensky,
2,
L. E. Dickson, Fir_s_t C o u r s e in the T h e o r y of Equations, John
McGraw-Hill,
1948,
Wiley, 1922,
xxxxxxxxxxxxxxxxxxxx
OMISSIONS •
H-3 and H-8 w e r e a l s o solved by John L. Brown, J r . , The P e n n s y l vania State University, State College, Penn.
The solution to H-16 given in the l a s t i s s u e was compounded from
solutions given by L. C a r l i t z and John L9 Brown, J r .
The v a r i t y p i s t
omitted the c r e d i t line,
H-13 was a l s o solved by John H. Halton,
Boulder,
U n i v e r s i t y of Colorado at
Colorado.
H-15 was a l s o solved by L. C a r l i t z , Duke University, D u r h a m , N. C.
ADVANCED PROBLEMS AND SOLUTIONS
Edited by VERNER E. HOGGATT, Jr.
San Jose State College, San Jose, California
Send all communications concerning Advanced Problems and
Solutions to Verner Ea Hoggatt, Jr. , Mathematics Department, San
Jose State College^ San Jose, California.
This department especially
welcomes problems believed to be new or extending old results.
Pro-
posers should submit solutions or other information that will assist
the editor.,
To facilitate their consideration, solutions should be sub-
mitted on separate signed .sheets within two months after publication
of the problems.
H-34
Proposed
by Paul F. Byrd, San Jose State College,
San Jose,
California
D e r i v e the s e r i e s expansions
sj0
9
4_1
s
J „ ( « ) = I, ( ° ) +
(-1)
I m+k
m= 1
(k = 0, I j 2, 3, , . . ) fortheBessel functions
where
H-35
m-k 1
J^,
2m
of all even orders,
L are Lucas numbers and 1 are modified Bessel functions.
n
n
Proposed
by Walter W. Horner, Pittsburgh,
Pa.
Select any nine consecutive terms of the Fibonacci, sequence and
form the magic squa.re
b
U
Generalize.
8U1U6
U
+U
3U5U7
+u
4u9a2 =
8U3U4 + U1U5U9 + U6U7U2
123
124
ADVANCED PROBLEMS AND SOLUTIONS
H-36
Proposed
by J.D.E.
Konhauser,
Consider a r e c t a n g l e R.
State College,
April
Pa.
F r o m the upper right c o r n e r of R r e -
move a r e c t a n g l e S ( s i m i l a r to R and with sides p a r a l l e l to the sides
of R.
D e t e r m i n e the l i n e a r r a t i o K = L 0 / L
if the c e n t r o i d of the
r e m a i n i n g L shaped region is w h e r e the lower left c o r n e r of the r e moved r e c t a n g l e w a s .
H-37
Proposed by H.W. Gould, West Virginia University,
Morganfown, V/est. Va.
Find a t r i a n g l e with sides n + l , n , n-1 having i n t e g r a l a r e a . The
f i r s t two e x a m p l e s a p p e a r to be 3, 4, 5 with a r e a 6; and 13, 14, 15 with
a r e a 84.
H-38
Proposed by R.G. Buschman, Suny, Buffalo,
N.Y.
(See Fibonacci N u m b e r s , Chebyshev P o l y n o m i a l s ,
Generaliza-
tions and Difference Equations Vol. 1, No. 4, Dec. 1963, pp.
1-7.)
Show
(u
, + s( - b ) r u
)/u =
v
n+r
n-r ' n
H-39
X
r
Proposed by Verner E. Hoggatt, Jr., San Jose State College, San Jose,
California
Solve the difference equation in closed f o r m
C
I O
n+Z
= C
,,+C
n+1
n
+ F
~
n+Z
j
where
C, = 1, C = Z, and F is the nth F i b o n a c c i n u m b e r .
1
Z
n
two s e p a r a t e c h a r a c t e r i z a t i o n s of t h e s e n u m b e r s .
H-40
Proposed by Walter Blumberg, New Hyde Park, L.I.,
Let U, V, A and B be i n t e g e r s ,
N.Y.
subject to the following con-
ditions
(i) U > 1 ,
(ii) (U, 3) = 1;
(iii) (A, V ) = l ;
(iv) V= V^- 1 )/ 5 •
Show A U+BV is not a s q u a r e .
Give
1964
ADVANCED PROBLEMS AND SOLUTIONS
125
SOLUTIONS
EXPANSIONS OF BESSEL FUNCTIONS IN TERMS OF
FIBONACCI NUMBERS
P-Z
Proposed
by P.F.
Byrd, San Jose State College,
San Jose,
California.
D e r i v e the s e r i e s expansions
j 0 ( « ) = t (-Dk £ ( . ) - ij +1 («)
F 2k+1 ,
k=0
where
J
and I
a r e B e s s e l functions, with F 9 1
,
being F i b o n a c c i
numbers.
Solution
by the proposer,
P.F.
Byrd, San Jose State College,
Note: This is a c o r r e c t e d v e r s i o n .
San Jose,
California
Initially this r e a d J ( x ) = . . .
which does not m a k e s e n s e .
It is j u s t as e a s y to d e r i v e the m o r e g e n e r a l s e r i e s expansions
of J ? ( a ) , (p=0, 1 , 2 , . . . ) , for B e s s e l functions of all even o r d e r s ,
Zp
and then to obtain the d e s i r e d r e s u l t as a s p e c i a l c a s e upon setting
p = 0.
We m a k e p r i n c i p a l u s e of f o r m u l a s (6. 1) and (6. 5) p r e s e n t e d
in [l] . Since
J ? (a) is an even function, we f i r s t seek a polynomial
expansion of the form
OO
J
2p(a)=k^0
? 2 k
»PM ^k+l(x)
'
w h e r e from equation (6.5) and [1] the coefficients a r e given by
7r
('\£,2k
5
2k, p ^
=
~
J
J 2 (-2i<* C osv) [cos2kv - cos(2k+2)v]
"'2p
0
being the F i b o n a c c i polynomials defined in [ 1 ] ,
with
_. ,,
Zk+1
known (e. g. , see [2]) that
dv ,
Now it is
7T
J
J \ (-2i« cosv) c o s 2 m v dv = 7T J ,
(~i«) J
(-i«) >
2p
p+m
p-m
0
and a l s o that J (iz) = i n I (-z) = ( - i ) n l (z). Hence we e a s i l y obtain
5
2k, P = ( - 1 ) P + k [ W t t ) I k - P ( a ) - W i ( t t ) W i ( 0 ) ]
•
126
ADVANCED PROBLEMS AND SOLUTIONS
Finally,
taking x = 1/2, and thus with <p
April
(1/2) = F . _, we have
the f o r m a l expansions
J
2
(«)=
P
E (-Dk+Prj
(«)I
(*)-I
(«)T
(all F •
k
k=0
' k+P
~P
k + p + l l M k-p+l v JJ ± 2k+L
which in p a r t i c u l a r yield the solution to the p r o p o s e d problem upon
setting p = 0*
REFERENCES
1.
, P. .F. Byrd, Expansion of Analytic Functions in Polynomials
A s s o c i a t e d -with F i b o n a c c i Numbers, The F i b o n a c c i Quarterly,,
Vol. 1, No,. 1,. pp. 16-29.
2.
G. N. Watson, A T r e a t i s e on the T h e o r y of B e s s e l Functions,
Cambridge, 2nd Edition, 1944, p a 151.
Note;
C o r r e c t e d statements, to P - l . '
Verify
that the polynomials
P
1
(x)
satisfy the differential
equation
•'
( l + x 2 , ) y " +' 3xy f - k(k+2)y = 0 ,
R e a d e r s a r e r e q u e s t e d t.o submit solutions to the problems in the above
mentioned r e f e r e n c e
[l]„
CORRECTIONS IN SAME P A P E R
P a g e 19 Replace 2 ] by [ 2 ] in line 1 1 .
3
P a g e 20 In line 5 r e a d (3) a s r e l a t i o n r e f e r r i n g to footnote 3 In view
of . . . .
'
P a g e 21 Read ^ . + 1 # x as y .(x) in (4. 7)
Page 23 In (5. 6) place absolute value b a r s around the quantity a p p r o a c h i n g the limit z e r o .
SYMBOLIC RELATIONS
H-18.
Proposed
by R..G. Buschman, Oregon State University,
"Symbolic relations
For example,
if
11
Corvaiiis,
Oregon
a r e s o m e t i m e s used to e x p r e s s identities.
F n • and L• n denote,
respectively,
.r
J
Fibonacci and
1964'
ADVANCED PROBLEMS AND SOLUTIONS
127
Lucas numbers,- then
<l:+.L)a=L^.
(l+F)n=F2n
are known identities* where = denotes that the exponents on the symbols are to be lowered to subscripts after the expansion is made,
(a) Prove
(L + F ) n = (2F) n .
(b) Evaluate
(L + L) .
(c) Evaluate
(F + F) . .
(d) How can this be suitably generalized?
Solution by the proposer (now at Sunyf Buffalo, N.Y.)
Consider the generating functions:
u/2
2 e ' H sinh{u./5~/2) = /s" £
n^O
u/2
2 e ' cosh(u/5/2) =
F
u /n!
n
,
•
1
z L u "/n.
n:
n=0
From these and the product formula for power series we can write
F L
(a)
/ 5 s F 2 u ' / /nl = 2 e sinh(ui/5)
=¥ i/5 £
£ Tt~"TTT7 u
x /
x v
*
0
ix
n
n=0
i
n
kf
n=0 k=0
(n-k):
'
'
n
/u\
(b)
^T
k n~l
n
^ u ,
,
/r,n
„
n
n
s
2 T T - r T T i u. = 2 e (cosh uy5 +1} = £• — - — — —
u .
v
i
n^O k~0
n=0
_
n F, F ,
o^> L 2 - 2 ^
k n-k
n
^ u ,
/=- , , /=- • _
n
n
y ,
1
(c)
x
£ 7T""T~~"rTT u ~ ^ e ' ( c o s h u i / b - i ) / 5 = £ — — - - — — u
n=0 k-0
'
•
,
.n=Q
Equating coefficients and multiplying by nj then gives
n
(a)
(b)
1
(c).
k=o © F* L - k
=
° r (F + L>n = (2F) '
" F-
n
£
( f ) L, L 1 = 2 n L + 2
, ~ \k/
ic n-k
n
k=u
n
s ( £ ) F k F n _ k = (2 n L n - . 2)/5
k=0
5o/uf/on &y L. Carlitzf Duke University,
' or
or
(L + L) n =' (2L) n + 2 ,
'
(F + F ) n =. (2L) n - 2)/5
Durham, NX.
As noted by Gould (this Quarterly, Vol. 1 (1963), p. 2) we have
128
ADVANCED PROBLEMS A N D SOLUTIONS
ax
e
bx
oo
n
^ x „
n
n n!
n= 0
- e
a-b
^
°1
A
n
x
ni
T
n=0
a =i(l +^,
It f o l l o w s a t o n c e t h a t
c^
,
bx
ax
April
b
= i(l
n
-f5)
2ax
2bx
n=0
so that
(L + F ) n = 2 n F
(a)
.
n
Similarly
„
£
x /r . T . n ,
2ax
—p ( L + L) = e
2ax
= e
. - (a+b)x , 2bx
+ 2ev
' + e
. 2bx . 0 x
+ e
+ 2e
,
so that
(L + L ) n = 2 n L
(b)
+ 2 ,
while
/
» \2
( « - 5)
so that
TT x .
.n
2ax
2bx
x
s - r r ( F + F) = e
+ e
- 2e
,
n=0 n '
5(F + F ) n = 2nL
(c)
'
- 2 .
n
consider
v
To g e n e r a l i z e t h e s e f o r m u l a s
, ax , bxxr
(e
+ e ) =
r
_
X
,r
(r-s)ax+sbx
( Jx e x
s=0
r
1
,r x s x . ( r - 2 s ) a x . ( r - 2 s ) b x .
;
j 2 (s) e
(e v
+ ev
)
s=0
,
2
Therefore
r
" V
s=0
°~
e
n
n=0
.
.n
nl
n
n *
n
1964
A D V A N C E D . F L O ; - «jy.i>/ir. A N D S O L U T I O N S
/ T J_T J
JLT
/*
li
,
,.
'-
,
5
--.
' ^
-
^
.k
n-k
129
T
^
r
k
r.,
T
7
>k
n-k
In p a r t i c u l a r
(L+L-l-J,)'"1 -- ^ L
SimilarLy-j
J3L,
since
5 r (r+... +F)n - i r (C \
. w h e r e the n u m b e r of P ! - »«= /
r
n
1
5r(F+. . . +F)n = T
'
^ i"i)sdr)(2r"2s)ksn~k .
j* H
, ^ ,
.
-S ?, y-fl
k n—k
T( \
)(2r-2s +l ) K s n K ,
L-d
w h e r e the n u m b e r of r 1 : JC ~ i 1
JU p a r t i c u l a r
A f o r m u l a for
,L +
s e
, -f L + F + . . . T F)
c a n be o b t a i n e d b u t i t i s ^ c r y c o m p l i c a t e d .
W h e n the n u m b e r of L ' s
f
i s e q u a l to the n u m b e r <yc F ;s i\' ic> l e s s c o m p l i c a t e d .
(L+L+F+F) 1 " 1 =r. i ; 4 n 1^ - 2 n + 1 )
In p a r t i c u l a r
.
Indeed s i n c e
/ a x , bx.r, ax
bx.r
,' 2 a x
2bx,r
(e
+ e ) ie
~ e ) = (e
- e
)
it f o l l o w s that
In p a r t i c u l a r
(L
+ L + L + F ! ^ + F
K
-= l ~- l (J F
"
D
n
- 3F, ) .
Z n ••
130
ADVANCED PROBLEMS AND SOLUTIONS
THE
H-1 9
Proposed
by Charles
April
RACE
R. Wall, Texas Christian
University,
Ft. Worth,
Texas
In the t r i a n g l e below [drawn for the c a s e ( 1 , 1 , 3 ) ] , the t r i s e c t o r s of angle, B, divide side, AC, into s e g m e n t s of length F , F
F
10.
n+3
,,
Find:
(i) lime
n-> oo
(ii) lim if
A
v\_
_/v_
n+3
"n+l
Solution
F
by Michael
Goldberg,
Washington,
D.C.
As n ^ o o , the r a t i o F , / F
a p p r o a c h e s t = (-v/5~+ l ) / 2 , a n d
, Q / E a p p r o a c h e s t ~ 2t + 1< Hence, the limiting t r i a n g l e ABC
can be drawn by taking points D and E on AC so that AD = 1, DE = t
3
and EC = t = 2t + 1. Since BD is a b i s e c t o r of angle ABE, the point
B m u s t lie on the c i r c l e which is the locus of points whose d i s t a n c e s
to A and E a r e in the r a t i o A D / D E = l / t . The c i r c l e p a s s e s through
D. If the d i a m e t e r of the c i r c l e is 2r = x + 1, then x / ( x + 1 + t) = l / t
from which
t/(t - 1) = t,2 = t + 1
1
Similarly,
BE is a b i s e c t o r of the angle DBG.
The point B m u s t lie
on a c i r c l e which is the locus of points whose d i s t a n c e s from D and C
a r e in the r a t i o D E / E C = t / t = l / t . If the d i a m e t e r of the c i r c l e is
2
2
2r^ = y + t, then y / ( y + t + t ) = l / t from which
1964
ADVANCED PROBLEMS AND SOLUTIONS
Hence,
131
cos < B A E = - t / 2 ( t + 1) = - {^5~- l ) / 4 and < B A E = 108° .
F r o m w h i c h 20 = 90° - 1 0 8 ° / 2 = 3 6 ° ; 6 = 1 8 ° , ^ = 180 ° - 108° - 3 6 = 1 8 °
Also solved by the proposer
and Raymond Whitney, Penn. State University,
Hazelton,
Penn.
FIBONACCI TO LUCAS
H-20
Proposed
San Jose,
by Werner E. Hoggatt,
California.
Jr., and Charles H. King,
San Jose State
College
If
snow
Q
1
D(e
)= e
0
w h e r e D(A) is the d e t e r m i n a n t of m a t r i x A and L
is the n
n
num be r .
Solution
by John L. Brown, Jr., Penn. StateUniversity,
State College,
R e c a l l that
F
Q
n =
F
F
n
so that (by definition)
2
.Q
F
n+1
nk+1
k.!
k=0
.
n-1
2
k=0
nk
^i
0s5
car
V
F
i ^n
-
k=0
n
n
k,
^
^
i ^n
F
nk-1
k=0—jr
Penn.
th
Lucas
A D V A N C E D P R O B L E M S AND SOLUTIONS
132
April
it i s w e l l - k n o w n t h a t
oo
J? ,nk
•v
—••
k=0 ~'
,
k
x
a x
b x
e
-e
^5
—•
—
[ e . g . e q u a t i o n ( 2 . 1 1 ) , p. 5 of G o u l d ' s p a p e r in V o l . 1, N o . 2, A p r i l
1963j , w h e r e
a
1_ +_V E_ T
=
and
,
1 - VT
Similarly,
L
,
• ,
n
n
u
nk • k
a x .
b x
k,
x = e
+ e
k=0
for the L u c a s n u m b e r s .
But
L , = F , , , + F , ,; therefore,
nk
nk+1
nk-1
°^
y
kt0
F
.
1 ,1 + F .
nk+1
nk-1
kl
, a
(e
2
nk+3.
k=0 "~k!
(1)
n
n
= e
,n
. b
+ e )
un
. b
+ e .
a
" • ^ -
2
k=0
Fnk+1 = F ^ + F ^
and
S
or
1
k!
i
•*-
00
Since
•
jr
00
^
-^-
= S
^ _
V^
k=0
from above, we a l s o have
00
/?\
F
^
k=0
1
, ,,
00
2bEZ± - -^
xv
k!'
k=0
k=0
F
,
nk
k
AVe
°
00
,
k=0
k=0
F
.
,
a
nk-1 _
x
*"'
e
e*o
b
- e
Y5~
,
k=0
S o l v i n g (1) a n d (2) s i m u l t a n e o u s l y , w e find
a
XT
nk+1
k!
(3)
k=0
1
+
( / * / )
e
(4)
k=0
Now,
D(e
Q-n
"nk^ 1 _ 1_
K
*
e
^
00
2
k=0
F ,
a
n
u
. b
+ e
n
/ °° F , ,
n
HK+1 \ I 2
nk-1
kl
M k=0" kl
. n
b
- e
V^r
and
T
n
a
e
n .
T
b
n
- e
i^
\
k=0
nk
kl
F
,
,
nk-1
k!
'
1964
ADVANCED PROBLEMS AND SOLUTIONS
e
a
n ltn
a +b
= e
H-21
n
,n \
+ eb >
/ a
I e
"
e
b
v
L
n
n
'
yr
9
.
T
since L
Proposed by Francis D. Parker, University
of Alaska, College,
,n
b
a
e
N
2
- e
f5
n, u n . .
= a +b for n > 0.
n
—
FIBONACCI PROBABILITY
= e
133
,
q. e„ d.
_±__
Alaska
th
F i n d the p r o b a b i l i t y , a s n a p p r o a c h e s infinity, t h a t t h e n" x F i b o n a c c i n u m b e r , F ( n ) , i s d i v i s i b l e by a n o t h e r F i b o n a c c i n u m b e r {/ F ,
orF2).
Solution by proposer
We u s e f r e q u e n t l y the f a c t that F(n) i s d i v i s i b l e by F(k) if k
d i v i d e s n.
T h e n the p r o b a b i l i t y that F(n) i s d i v i s i b l e by 2 i s 1 / 3 ;
1 2
the p r o b a b i l i t y that F(n) i s d i v i s i b l e by 3 b u t n o t 2 i s (^)(4); t h e p r o b ~~~"
" " 1 2 3
2
a b i l i t y that F(n) i s d i v i s i b l e by 5 but not by 2 or by 3 i s •=• -^ T = ^ | f ;
and i n g e n e r a l the p r o b a b i l i t y that F(n) i s d i v i s i b l e by F(k) but n o t any
F i b o n a c c i n u m b e r of o r d e r l e s s than k i s T T T T ^ - T h e s e p r o b a b i l i t i e s
k(k-l)
a r e a l l i n d e p e n d e n t , s o that the p r o b a b i l i t i e s that F ( n ) i s d i v i s i b l e b y
at l e a s t one F i b o n a c c i n u m b e r of o r d e r not e x c e e d i n g k i s
1
Z
?.
?.
3.2 + T
4 .~ 3: + 5 . 4
' ' ' k(k-l) "
k-2
T h i s s u m i s —-:—, and a s n a p p r o a c h e s infinity, the p r o b a b i l i t y a p K.
p r o a c h e s unity.
A l s o s o l v e d by J. L. B r o w n , J r . , P e n n . State U n i v e r s i t y , , State
College,
Pennsylvania.
134
April
Continued from P a g e 114
k = F , , , we have that k+1 = F ., + F „ , and we a r e through.
If
n+1
n+1
2
k = F ,, - 1, we have k+1 = F , , and we a r e through.
If k = F ,-, - 2
&
n+1
n+1
n+1
we have k+1 = F , - 1, which by L e m m a 1 (A or B) ? can be r e p r e sented as c l a i m e d and we a r e through again. T h e r e f o r e let us cons i d e r k i F ., - 3.
n+i
Now the r e p r e s e n t a t i o n f o r k in this form can best be e x p r e s s e d
a s k = F +a
~ F » +a - F „ +a
. F
. + . . . + a^F- + a 0 F n
n - 2 n-2
n-3 n-3
n-4 n-4
3 3
2 2
w h e r e a. = 0 or 1 for 2 £. i S n - 2 , and a. = 1, i m p l i e s that a. ± 1 = 0.
i
I
x
i
Now t h e r e a r e only two p o s s i b i l i t i e s for a ? and a, in this r e p r e s e n t a tion.
E i t h e r a ? = a . = 0, or a
^ a«.
If the f i r s t c a s e is t r u e for k,
we can r e p r e s e n t k+1 in the r e q u i r e d m a n n e r , simply by adding 1 to k
in the form of a ? = 1. If the second c a s e is t r u e for k, we then c l a i m
that t h e r e e x i s t s at least one place in the r e p r e s e n t a t i o n w h e r e a. ='
a. . , = 0 , since o t h e r w i s e , k = F ., - 1 which we have a l r e a d y taken
l+l
n+1
. '
c a r e of above.
T h e r e f o r e we can r e p r e s e n t k+1 by the following:
k+1 = F +a ~F „ + . . . + a . , - F . ,., + a.' F . , +. . . + a 'F 0 + a 0 F 0 +1
n
n-2 n-2
i+2 i+2
i-l i-l
3 3
2 2
Now c o n s i d e r the e x rp r e s s i o n from a. , n F . ._ on and the r e s u l t i n gto in-'
i+2 1+2
equality.
a i+2
. i 7 F .i+2
. , - - 1 <. F n - 1, - 1,
l7+a,
i - l. F .i - l, + . . . + a 3Q F 3. + a 02F „2 +1 < F i+3
by our Inductive A s s u m p t i o n , Also by the Inductive A s s u m p t i o n , we
can r e p r e s e n t the e x p r e s s i o n from a. , ~ F . ~ on in the p r o p e r form
which i m p l i e s that we can then a l s o r e p r e s e n t k+1 in the p r o p e r f o r m .
This shows that the proof holds for all positive i n t e g e r s N. Q. E. D.
P h i l Lafer, Oak H a r b o r , Ohio
BEGINNERS' CORNER
Edited by DMITRI THORO
San Jose State College
T H E E U C L I D E A N A L G O R I T H M II
1.
INTRODUCTION
In P a r t I [ l ] w e s a w t h a t t h e g r e a t e s t c o m m o n d i v i s o r
numbers
could be c o n v e n i e n t l y c o m p u t e d via the f a m o u s
algorithm.
Suppose that exactly
c o m p u t e t h e g. c. d. of
s
and
n
t
We t h e n h a v e
(1)
s = t q , + r ,
0 < r
< t
(2)
t = r , q^ + r ,
0 < r?
< r
0 < ^
< rz
0 <
< r^
rx = r 2 q
(4)
r2 = r3 q4 + r ^
(5)
r 3 = r 4 q5 + r ^
(n-1)
N
'
r
(n)
x
'
r
0
n-3
= r
3
+ xy
^ q
-,+r
n-2 ^n-1
r
r
l-^
2
r
>r
3
r
3
> r
etc,
135
+ r
2
+
4
3
r 4
+ r
5
x
4
, < r
9
n-1
n-2
.
t > r: + r2
(3')
(5')
0<r
1
n-1,
5
q. > 1, t h e a b o v e e q u a t i o n s i m p l y
(2 1 )
(4')
T^
Q < x
~ = r
, q + 0
n-2
n - 1 ^n
Since each quotient
Euclidean
s t e p s ( d i v i s i o n s ) a r e r e q u i r e d to
(s > t ) .
(3)
of . t w o
136
F'F.CJNr^Rt' ^ ' r i i orw
F r o m (21) and ^ i,
(2 r . + 2 r J + r_,
2 r . , etc.
-„ ~, * -' ,• , ' /, I r o m (4 f ) ?
r
Sim:^"Ly, L - -.. t'i \
-1
Continuing In ch. s JI* u % . .v
of Fibonacci n u m b e r s .
April
2 r
+ r
>
^ -! " r . > (3 r . + 3 r . ) +
j
i<^ * he g e n e r o u s abundance
Thus
t •£• r , + r > 2r -1-r, > 3r n -f2r, > 5r ,+3r r
1 2
2 j ~~
.:>
4 *~
4
5
> ¥ , r _ +F n r _
"~ n-1 n - 2
n-2 n-1
2. A BASIC RESULT
Since the r e m a i n d e r s form a sti'i^tiy d e c r e a s i n g sequence with
r
, the last n o n - z e r o r e m a i n d e r .
n-1
c
„ v r
XI-&
"*
11- I
Consequently,
n-1
xi" L
n-L
) j - ! '""
n- 1
n-Z
n+1
To s u m m a r i z e , if n divisions a r e r e q u i r e d to compute the g. c. d. of
st
s and t, then t is at l e a s t as l a r g e as the (n + 1)
Fibonacci
number!
3. LAME'S THEOREM
Although the Euclidean a l g o r i t h m is over 2,000 y e a r s old, the
following r e s u l t was e s t a b l i s h e d by G a b r i e l Lame in 1844.
Theorem
The n u m b e r of divisions r e q u i r e d to find the g„ c. d. of two n u m b e r s is n e v e r g r e a t e r than five t i m e s the n u m b e r of digits in the s m a l l e r
number.
Proof.
Let
4> designate the golden r a t i o .
*pn = F n <t> + F n ^ | ,
In [2] it was shown that
n = l , 2, 3, . . .
1964
BEGINNERS' CORNER
Now since 2 > <t> = (1 + ^/5)/Z9
2F
+ F
n
137
we see that
, > F
ii-1
4> + F
,
or
n-1
n
> *n
F
n+2
Replacing n by n~l
section yields
and using the "basic r e s u l t " of the p r e c e d i n g
t > *n-1
.
To complete the proof note that
(i)
if t has
(ii)
log t > (n-1)
(iii)
Thus d > ( n - l ) / 5
d digits then d > log t
log 0
log 4> > 1/5 • .
or
n < 5d .
REFERENCES
1.
D. E. Thoro, "The E u c l i d e a n Algorithm I, " F i b o n a c c i Q u a r t e r l y ,
Vol. 2, No. 1, F e b r u a r y 1964.
(Note
that i n e x c e r c i s e s E8 and E10,
v
should be r e p l a c e d byJ
^
respectively. )
2.
D. E. Thoro,
N(F , . ,
n+1
"The Golden Ratio:
tions, " Fibonacci Quarterly,
(F . , , F ) and m a x (n, F - l )
n+1
n
n
F ) and m a x N (n, F - l )
n'
n
Computational
Vol. 1,
pp. 5 3 - 5 9 .
xxxxxxxxxxxxxxxxxxxx
No. 3,
Considera-
October 1963,
EXPLORING FIBONACCI NUMBERS WITH A CALCULATOR
BROTHER U. A L F R E D
St. Mary's College, C a l i f o r n i a
It h a s often been noted that the study of n u m b e r s is both e x p e r i m e n t a l and t h e o r e t i c a l in c h a r a c t e r .
culator,
Even in the days before the c a l -
to say nothing of the c o m p u t e r ,
the t r u l y g r e a t m a t h e m a t i -
cians often a r r i v e d at beautiful r e s u l t s on the b a s i s of o b s e r v a t i o n
and n u m e r i c a l w o r k before they p r o c e e d e d to proof and t h e o r e t i c a l
justification.
If we find o u r s e l v e s enjoying calculation and seeing
tangible r e s u l t s , we a r e in v e r y good company and need not w o r r y
about the attitude of the t h e o r i s t who is afraid to soil his hands with
n u m b e r sa
In this vein,
the following exploration is p r o p o s e d .
It m a y be
o b s e r v e d by looking at a list of F i b o n a c c i n u m b e r s that in c e r t a i n c a s e s
F h a s the n c o r r e s p o n d i n g to the t e r m i n a l digits of the n u m b e r .
n
°
.
°
Thus F 5 is 5; F
is 514229; F 6 l is 2504730781961. As long as
we have a table of Fibonacci n u m b e r s on hand we can p r o c e e d to m a k e
such v e r i f i c a t i o n s .
But suppose we set out to find all t h e s e coinci-
d e n c e s for F i b o n a c c i n u m b e r s up to a c e r t a i n level such a s
F,
n
nrm-
In the a b s e n c e of t h e s e n u m b e r s we now have an i n t e r e s t i n g m a t h e m a t i c a l p r o b l e m involving computation,
One v e r y simple way to p r o c e e d would be to take the s u c c e s s i v e
F i b o n a c c i n u m b e r s modulo 10, 000. In. other w o r d s we would c o n s i d e r
only the last five digits and forget about all those that go before.
This
is a s t r a i g h t f o r w a r d p r o c e d u r e but it would be long and tedious and
subject to e r r o r .
In fact,
once a m i s t a k e in introduced, a l l r e s u l t s
t h e r e a f t e r would be vitiated.
there are several ways.
T h e r e m u s t be a b e t t e r way.
Perhaps
We shall look f o r w a r d to both the n u m e r i c a l
r e s u l t s and the method employed in a r r i v i n g at t h e m .
A d d r e s s ail c o m m u n i c a t i o n s r e g a r d i n g this p r o b l e m to:
U. Alfred, St. M a r y ' s College, California.
in the i s s u e of D e c e m b e r , 1964.
138
Brother
The solution will a p p e a r
AMATEUR INTERESTS IN THE FIBONACCI SERIES-PRIME NUMBERS
JOSEPH MANDELSON
U.S. Army Edgewood Arsenal, Maryland
My i n t e r e s t in the Fibonacci s e r i e s was born in 1959 when it
was noticed that the p r e f e r r e d r a t i o s developed in the r e s e a r c h of m y
colleague, H. E l l n e r ,
and l a t e r included in D e p a r t m e n t of Defense
Handbook HI0 9 [ 1 ] , w e r e 1, 2, 3, 5 and 8.
F r o m r e c o l l e c t i o n of a
brief mention in college a l g e b r a , this was recognized as the f i r s t few
t e r m s of the F i b o n a c c i .
To t e s t the supposition that the p r e f e r r e d
r a t i o s would all be from this s e r i e s , the next one was calculated and,
s u r e enough, it was 13.
Then it was noted that the s a m p l e s i z e s ,
Acceptable Quality Levels (AQL's) and lot size r a n g e s of all sampling
s t a n d a r d s since Dodge and Rornig [2] w e r e s e r i e s a p p r o x i m a t e l y of
the type:
(1)
u
^
n+^
~- u
+u
n+1
n
In fact the l a t e s t v e r s i o n of M i l i t a r y Standard Mil Std 105 [3] shows
s a m p l e s i z e s which a r e a l m o s t exactly the Fibonacci s e r i e s itself.
These o c c u r r e n c e s w e r e too r e m a r k a b l e to be a s c r i b e d to m e r e c o incidence and my i n t e r e s t led m e to examine the s e r i e s e m p i r i c a l l y .
A c c o r d i n g to Dickson [4], the l i t e r a t u r e on this subject is r i c h , extending a s it does from the y e a r 1202 to the p r e s e n t .
However, it is
a l m o s t completely unavailable to m e and, I suspect, to m o s t o t h e r s .
On developing the s e r i e s
u
from
n = 0 to n - 25 or so, in-
spection soon r e v e a l e d that two t h i r d s of the s e r i e s c o m p r i s e d odd
n u m b e r s and exactly e v e r y t h i r d u
was even.
It did not take much
to a s c e r t a i n why this is so. In this way I found that n, the ordinal of
u
in the s e r i e s w a s , in a m a n n e r of speaking, the d e t e r m i n a n t of the
p r o p e r t i e s of u .
Thus, if z is a factor of u
a factor of u ? , u„ , e t c .
it will infallibly be
T h e r e f o r e , in general* if n is c o m p o s i t e ,
so is u n x(except
^ for the c a s e n• = 4. un = 3),
' but if n is ^p r i m e , un
m a y be p r i m e . My f i r s t g u e s s that, since the density of odd n u m b e r s
139
140
AMATEUR INTERESTS IN THE FIBONACCI
in the Fibonacci is twice that of the even numbers,
April
the density of
primes would be greater than in the cardinal number domain was proven wrong when the primality of u
prime.
was found dependent on n being
The next supposition of equal density was shown to be wrong
when u~, = 1346269 was found to be a composite of 557 and Z417.
When u^ 7
and u . ,
were also determined to be composite it became
obvious that the density of primes in u was less than that of the
3
r
n
cardinal domain.
Several other interesting details were elucidated after extending
and examining the series,
finally to n = 130.
first down to n = 50 then to n = 100 and
No u
is divisible by n except when n ~ 5 or
powers of 5. For example uR = 5 and \i?i- - 75025. Except for U/,
every u
seem>s to have at least one prime factor which has not been
J
n
a factor of any previous
factors.
u ; some have two or three such new prime
Surely, any theory of prime numbers might profit from Fib-
onacci considerations.
However, the first gain from the extension of study of the series to n = 100 was a remarkable regularity found from the fact that
if P
is a prime factor of u
it will also factor, more generally,
to
J
n
^
n
u.
where j goes from 1 to 00. Consider the multiple j and let this
be expressed as a sum of multiples of powers of P , reduced to a
minimum of terms, and provided that no multiples of the powers of
P
n
> P .
n
Thus:
(2)
x
'
j = aP° + b P 1 + c P 2 + . . . qP1"
n
n
n
^ ii
J
where a, b, c . . . q may be zero but must always be less than P .
Then u.
will be divisible by
P x + 1 where P
is the lowest xpower
J
jn
n
n
term of P
in the sum of multiples of powers of P = i.
n
r
r
II J
Example 1.
The first prime to divide u
is 2 (P = 2) and it divides the
r
N
n
n
'
third number
we have:
(n = 3) in the series: U3 = 2.
From the above lemma
1964
SERIES -- PRIME ]NUMBERS
S u m of P
Ordinal
*
p
n
x
jn
J
terms = j
3
1
Pn
0
6
2
P1
n
1
^0
9
P
24
8
P3
30
10
33
11
^1
Pn°
P
3
^3
+ Pn
n
^3
P n J_
+ p + Pn
r.0
n
p
is divisible
by P x
J
0 n
+
P
nl +
P
n
0
+ Pn
n
n
Since P
_L
jn
X
0
3
u.
n
3
+1
P
0
P°
n
u.
1
1
= Pn = 2
1
= P
2
2
= 4
n
8
+ 1 _ PX = 2
n
+ 1 P 4 , 16
=
n1 + 1
1
141
+1
=
=
n
P
n2
34
46368
= 4
332040
PX = 2
3524578
n
n
= 2, no m u l t i p l e s other than 0 or 1 a p p e a r in the sum of
p o w e r s of P
= j . Actually the sum of m u l t i p l e s of power t e r m s for
j = 11 should read:
11 = I P
0
1
2
^
E x a m p l e 2.
Another p r i m e dividing u
tioned,
5
n
j terms = j
1
P°
P
n
X
0
n
10
2
2P°*
n
0
20
4
4P°n
0
25
5 P1
30
n
6 P°
n
35
1
^
is 5 (P
0
+2
1
+ 2
3
7
1
+ P1
2P°n
50
10 2 P
25
25 P
2
n
1
n
+
n
P1
n
0
0
1
2
= 1 + 2 + 8.
= 5) and, a s a l r e a d y m e n -
it divides the fifth n u m b e r in the s e r i e s : u_ = 5.
mcike a table:
Ordinal
Sum of P
Jn
0
+ I P + OP + I P = P + P + P = 2
n '
n
n
n
n
n
n
Again we
u. is divisible
jn
by P_ x+1
J
n
u.
p 0 + l
= P1 = 5
5
n
n
1
p o + i -P
55
= 5
n
n
p0+l ^P1
6765
= 5
n
n
1+1
75025
P
= P 2 = 25
n
n
P o+i ^ P 1 = 5
832040
n
n
p 0 + l = PX
9227465
= 5
n
n
1+1
12586269025
P
= P 2 = 25
n3
n 2+1
P
= P = 125
n 59425114757512643212875125
n
142
A M A T E U R I N T E R E S T S IN T H E F I B O N A C C I
*
April
0
T h e m u l t i p l e Z of 2 P
in this c a s e ) is u s e d .
At a l a t e r t i m e ,
recommended
p l a y s no parts
o n l y t h e p o w e r of
in a p r i v a t e c o m m u n i c a t i o n *
D i c k s o n [4] a s a r e f e r e n c e
a c c o r d i n g to
Dickson,
to t h e l i t e r a t u r e .
the above w a s
T h e o r e m V of e i g h t i n t h e f o l l o w i n g
"If
n
(zero
D r . S« M .
I d i s c o v e r e d that t h e s e findings w e r e known to L u c a s [ 5 ] .
ticular,
P
stated by
Ulam
In t h i s
In
par-
Lucas
as
form:
i s t h e r a n k of t h e f i r s t t e r m
u
containing the p r i m e
n
&
ris the f i r s t t e r m divisible by
f a c t o r p to t h e p o w e r A, t h e n u
X+l
X+2
^ >n
p
a n d n o t by p
; t h i s i s c a l l e d t h e l a w of r e p e t i t i o n of p r i m e s
in t h e r e c u r r i n g to; s e r i e s of u . "
n
On r e a d i n g t h i s it i s c l e a r t h a t p r e c e d e n c e i n t h i s f i n d i n g l a y
with Lucas who had,
ically.
moreover,
s t a t e d it m o r e c l e a r l y and e c o n o m -
F a r from being discouraged, however, I continued m y s e a r c h ,
l i s t i n g all p r i m e n u m b e r s up to 10009 and l a b o r i o u s l y t e s t i n g the p r i m a l i t y of m o s t
!
u
s
u p to
n = 130.
Of c o u r s e ,
p r i m e s up to 10009
a r e s u f f i c i e n t o n l y to t e s t u
u p to n = 40 d i r e c t l y b u t t h e f a c t t h a t
if z d i v i d e s u
it w i l l d i v i d e u .
h e l p e d ag r e a t l yJ . N e v e r t h e l e s s i t
^
jii
n
•
s p e e d i l y b e c a m e a p p a r e n t t h a t r e p e a t e d d i v i s i o n of u
g r e a t e r than
u . j - on a d e s k c a l c u l a t o r w a s n o t o n l y l a b o r i o u s b u t i n c r e a s i n g l y p r o n e
t o e r r o r a s t h e n u m b e r of d i g i t s in u
r o s e a b o v e 1 0 . If o n l y t h e r e
J
°
n
w e r e s o m e w a y to e l i m i n a t e s o m e of t h e t r i a l d i v i s i o n s !
A s t u d y7 of 1t h e p r i m e s , P , w h i c h d i v i d e u
r e v e a l e d t h a t t h eJy
n
n
w e r e a l l of t h e f o r m
(3)
P n = an + 1
Since
P
i s p r i m e it i s o b v i o u s t h a t a n h a d to b e e v e n s o t h a t a n ± 1
n
could be odd,
T h e r e f o r e e i t h e r a o r n o r b o t h h a d to be e v e n .
C l o s e r s t u d y of t h e p r i m e s i n d i c a t e d t h a t , w h e n
e v e n , it w a s a l w a y s n e c e s s a r y to a d d o n e to
a
and
n
even,
P
= a n + 1,
never
a n - 1.
an
a
and
n w e r e both
to g e t
P , i . e. w i t h
I c a n n o t e x p l a i n t h i s but,
e m p i r i c a l l y , i t t u r n s o u t t h i s w a y . Now i t w a s p o s s i b l e t o c u t d o w n o n
t h e n u m b e r of d i v i s i o n s r e q u i r e d to d e t e r m i n e t h e P
which would
•L
n
1964
SERIES — PRIME NUMBERS
divide u n .
Thus:
a.
Calculate
b.
D e t e r m i n e whether
c.
Divide u
d0
If u
2n + 1 (If n is even, d e t e r m i n e only 2n + 1).
2n + 1 a n d / o r
2n - 1 a r e p r i m e .
by any p r i m e n u m b e r d e t e r m i n e d in a and b.
is not divided in c. calculate
n
Repeat setps c. and d,
e„
3n + 1.
f. If u
is not divided in step e, calculate
n
even, d e t e r m i n e 4n + 1 only).
e.
&
143
4n + 1
If n is
Continue until the P
which divides u is found.
n
n
The r e l a t i o n s h i p found above m a y be e x p r e s s e d a s follows;
If P is any p r i m e t h e r e e x i s t s an n such that P
an - 1 will divide u
b e r > 0. )•
without r e m a i n d e r (a being s o m e whole n u m -
The only exception is
/
J
= an + 1 or
1
P
=5
which divides u c = 5.
5
n
It is p o s s i b l e that the above r e l a t i o n s h i p would r e p a y i n v e s t i g a tion in p r i m e n u m b e r t h e o r y . In the past, a n u m b e r of f o r m u l a s have
been p r o p o s e d for the p u r p o s e of generating p r i m e n u m b e r s . In e v e r y
c a s e the f o r m u l a s have been found faulty in one or m o r e of the following r e s p e c t s :
a.
The density of p r i m e s gene r a t ed lias been much lower than
the t r u e density of p r i m e s .
b.
They have g e n e r a t e d c o m p o s i t e n u m b e r s ,
c.
They have r a r e l y been capable of g e n e r a t i n g p a i r e d p r i m e s
(two consecutive p r i m e s which differ by 2, e. g'. 11 and 13).
The formula given in (3) suffers only in generating v e r y m a n y c o m p o s i t e s . However, the p r o c e d u r e c l e a r l y furnishes a c r i t e r i o n w h e r e by (empirically) it h a s been found that,
divided by P
if n is p r i m e ,
u
will be
only when p , determined, as in (3), is p r i m e .
If
this can be proven, new light m a y b e shed by the proof on this a g e - o l d
problem.
144
AMATEUR INTERESTS IN THE FIBONACCI
April
REFERENCES
1.
D e p a r t m e n t of Defense Handbook HI09 "Statistical P r o c e d u r e s
for D e t e r m i n i n g Validity of S u p p l i e r s ' A t t r i b u t e s Inspection, "
6 May I960.
2.
Dodge and Romig, "Sampling Inspection Tables, " Second E d i tion, 1959, John Wiley and Sons, Inc.
3.
M i l i t a r y Standard Mil Std 105D,
"Sampling P r o c e d u r e s and
Tables for Inspection by A t t r i b u t e s , " in p r o c e s s of publication
by D e p a r t m e n t of Defense.
Revision Mil Std 105C is dated 18
July 1961.
4.
Dickson,
" H i s t o r y of the Theory of N u m b e r s , " Chapter XVII,
5.
R e c u r r i n g S e r i e s , Lucas 1 u , v .
°
-n
n
Lucas, "Sur la the'orie d e s n o m b r e s p r e m i e r s , " Atti R. Accad.
Sc. Torino (Math. ), 11, 1875-6, 928-937, cited in [ 4 ] .
XXXXXXXXXKXXXXXXXXXX
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THREE WEEKS PRIOR to publication
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RENEW YOUR SUBSCRIPTION! I
A MOTIVATION FOR CONTINUED FRACTIONS
A.P. HILLMAN and G.L. ALEXANDERSON
University of Santa Clara, Santa Clara, California
This Q u a r t e r l y is devoted to the studyof p r o p e r t i e s of i n t e g e r s ,
e s p e c i a l l y to the study of r e c u r r e n t s e q u e n c e s of i n t e g e r s .
We show
below how such s e q u e n c e s and continued fractions a r i s e n a t u r a l l y in
the p r o b l e m of a p p r o x i m a t i n g an i r r a t i o n a l n u m b e r to any d e s i r e d
c l o s e n e s s by r a t i o n a l n u m b e r s .
We begin with the equation
x2 - x - 1 = 0 .
(1)
One can e a s i l y see that t h e r e is a negative root between -1 and 0 and
2
a p o s i t i v e root between 1 and 2, for e x a m p l e by graphing y = x - x - 1.
We call the positive root r .
This n u m b e r has been known since a n -
tiquity a s the "golden m e a n . " We now look for a sequence of r a t i o n a l
a p p r o x i m a t i o n s to
r.
A r a t i o n a l n u m b e r is of the form
(and q ^ 0).
p / q with p and q i n t e g e r s
We t h e r e f o r e wish two s e q u e n c e s
p
l*
q
r
P
2'
P
3S ' * '
(2)
q s
z
q s
3
of i n t e g e r s such that the quotients p / q
get a r b i t r a r i l y c l o s e to r .
a r e a p p r o x i m a t i o n s which
It would a l s o be helpful if each new a p -
p r o x i m a t i o n w e r e obtainable s i m p l y from p r e v i o u s o n e s .
We go back to equation (1) and r e w r i t e it as
(3)
\
x = 1 +1 .
J
x
This s t a t e s that if we r e p l a c e x by r
(4)
x
'
1 + x
145
in
146
A MOTIVATION FOR CONTINUED FRACTIONS
the result Is r
April
and suggests that if we replace x in (4) by an approx-
imation to r we will get another approximation.
We now change (3)
into the form
and consider
of l / x ,
x,
to be an approximation to r.
is the same as that of x,
tive e r r o r of x ?
(ic e. j
and? if x,
1 + 1 /x ) is lower than that of x , ,
adding 1 increases the number but not the e r r o r .
x?
in (5) is a better approximation to r
since
It can be shown that
that x , if x
We now let our first approximation x
Pi Ah
The relative e r r o r
is positive, the rela-
> 0.
be a rational number
and substitute this in (5) obtaining
*i
Pi + ^
= 1 + ——;r— = 1 + -— =
x
^pi/qr
*
We therefore choose p ?
to be p, + q,
our third approximation is
general, the
(n + l ) - s t
p
p /q„
p
i
and q ? to be p, .
r
w ith p
=p
approximation p
(6)
Pn+l
= P
n
<7>
VU
= P
n•
It follows from (7) that q
=p
(8)
%
'
+ q
r
,,
and q„ = p . In
has
, ; substituting this in (6) gives
.
is between 1 and 2 we use 1 as the first approximation,
i. e. , we let p. = q, = 1.
from (8) that p
, /q
-f q
Similarly,
n
p ,, = p + p
^n+1
n
^n-l
Since
.
i
This means that p ? = 2 and it now follows
is the Fibonacci number
F ,1 -
Then (7) implies
that q = F
and we see. that the sequence of quotients F , , / F of
^•n
n
T.
A
n+1 n
consecutive Fibonacci numbers furnishes the desired approximations
to the root r of (1). It can be shown that this sequence converges to
r in the calculus sense.
1964
way.
A MOTIVATION FOR CONTINUED FRACTIONS
We next c o n s i d e r the p r o b l e m of a p p r o x i m a t i n g
The n u m b e r s is the positive root of
147
s = vTO in this
x 2 - 10 = 0 .
(9)
We w r i t e (9) in the f o r m s
2
9 =1
(x - 3)(x + 3) = 1
(10)
( x - 3) = l / ( x + 3)
x = 3 + l / ( x + 3)
and change (10) into
l
*u-=3+
n+1
3 -I- x
(11)
Again,, if x
x
is a positive a p p r o x i m a t i o n to s s it can be seen that
, is an a p p r o x i m a t i o n with s m a l l e r r e l a t i v e e r r o r .
sequence of r a t i o n a l a p p r o x i m a t i o n s
p ... = 3p + lOq ,
^n+1
^n
^n
p /q
T h e r e is a
with
q 1 = p + 3q
^n+1 *n
^n
Letting the f i r s t a p p r o x i m a t i o n be 3 5 i. e. , letting p, = 3 and q
we obtain the sequence
3 / 1 , 1 9 / 6 , 117/37, ... .
which can be shown to converge to s0
Equation (11) contains the equations
1
X
2
= 3
+
3 + xx '
1
x
3
+
~3 + x 2
Substituting the f i r s t of t h e s e into the second gives us
= 1,
148
A MOTIVATION FOR CONTINUED FRACTIONS
*3
= 3
+
April
*1
6+3 +
X;L
If this is substituted into x 4 = 3 + 1/(3 + x ) and if we let x,
be 3,
we obtain
l
x, = 3 +
In this way we
the x .
n
can write continued fraction expressions for any one of
Then it is natural to let the infinite continued fraction
3+-
1
6+
represent the limit
l
6 +. ..
s of the sequences
x
defined by (11) and
The infinite continued fraction for the root
r
ofx
X]
= 3.
- x - 1 - 0
1+
i
+
1
1 +...
whose elegant simplicity is worthy of the title "golden mean. "
xxxxxxxxxxxxxxxxxxxx
FIBONACCI AND LUCAS NUMBER TABLES
Those interested may secure bound mimeographed tables of the
first 1505 Fibonacci numbers, F , and the first 1506 Lucas numbers,
L , by sending two dollars to Professor Jack K. Ward, Westminster
College, Fulton, Missouri.
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900
MAXEY BROOKE
Sweeny, Texas
In 1202, a r e m a r k a b l e m a n w r o t e a r e m a r k a b l e book.
The
m a n w a s L e o n a r d o of P i s a , known a s Fibonacci, a b r i l l i a n t m a n in an
intellectual wilderness.
The book Liber Abacci (The Book of the
Abacus) introduced A r a b i c n u m b e r s into E u r o p e .
In the book w a s a s e e m i n g l y s i m p l e little p r o b l e m :
"A p a i r of r a b b i t s a r e enclosed on. all s i d e s by a wall.
To find
out how m a n y p a i r s of r a b b i t s will be born in the c o u r s e of one y e a r ,
it being a s s u m e d that e v e r y month a p a i r of r a b b i t s will p r o d u c e
another pair,
and that r a b b i t s begin to b e a r young two months after
t h e i r own b i r t h . "
On the m a r g i n of the m a n u s c r i p t , F i b o n a c c i gives the tabulation:
A pair
1
First
2
Second
3
Third
5
Fourth
8
Fifth
13
Sixth
21
Seventh
34
Eighth
55
Ninth
89
Tenth
144
Eleventh
233
Twelfth
377
He s u m s up h i s calculations
149
150
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 A p r i l
". - . w e see how we a r r i v e at it.
We add to the f i r s t n u m b e r
the second one* i . e M 1 and 2; the second to the third; the t h i r d to the
fourth; the fourth to the fifth; and in this way, one after a n o t h e r , until
we add together the tenth and eleventh and obtain the total n u m b e r of
r a b b i t s — 377; and i t i s p o s s i b l e to do this in this o r d e r for an infinite
n u m b e r of m o n t h s . "
T h e r e the m a t t e r lay for 400 y e a r s .
[ 1 ] of a s t r o n o m y fame,
21, . . .
In 1611, Johann Kepler
a r r i v e d at the s e r i e s 1, 1, 2, 3, 5, 8, 13,
T h e r e is no indication that he had a c c e s s to one of F i b o n a c c i ' s
h a n d - w r i t t e n books (The Liber Abacci was not published until 1857 [ 2 ] ) .
At any r a t e , in d i s c u s s i n g the Golden Section and phyiiotaxis, Kepler
wrote:
" F o r we will always have a s 5 is to 8 so is 8 to-13, p r a c t i c a l l y ,
and a s 8 is to 13, so is 13 to 21 a l m o s t . I think that the s e m i n a l f a c culty i s developed in a way analogous to this p r o p o r t i o n which p e r p e t u a t e s itself, and. so in the flower is displayed a pentagonal standard^
so to speak.
I let p a s s all other c o n s i d e r a t i o n s which might be a d -
duced by the m o s t delightful study to e s t a b l i s h this t r u t h .
,f
Simon Stevens (1548-1620) a l s o wrote on the Golden Section.
The editor of his w o r k s , A. G e r a r d [3] a r r i v e d at the formula for
e x p r e s s i n g the s e r i e s in 1634
Un+£
, o = Un+1
,, + Un
A hundred y e a r s m u s t p a s s before the p r o b l e m is again considered.
In 1753,
R. Simpson [4] d e r i v e d a formula,
implied by
Kepler
u
, u ,, - u 2 = (-i) n + 1
n-1 n+1
n
'
A s e c o n d h u n d r e d y e a r s p a s s by and the s e r i e s again c o m e s under
study.
In 1843, J. P . M. Binet [5] d e r i v e s an a n a l y t i c a l function for
d e t e r m i n i n g the value of any F i b o n a c c i n u m b e r
2 n ^ 5 U a = (l + ^ / 5 ) n - ( I -
sf5)n
1964 FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 151
The following y e a r , B. Lame [6] f i r s t used the s e r i e s to solve
a p r o b l e m in T h e o r y of N u m b e r s . He investigated the n u m b e r of ope r a t i o n s needed to find the GCD of two i n t e g e r s (it does not exceed 5
t i m e s the n u m b e r of digits in the s m a l l e r n u m b e r ) .
Two y e a r s l a t e r , E. Catalan [7.] derived the i m p o r t a n t formula
n-1
L
_n
un -
T
5 n ( n - l ) ( n - 2 ) , 5 2 n ( n - l )(n-2)(n- 3)(n-4) J
+ - - _ - - _ _ +_^™ T ___ r i T ____„ + . . .
By now, the s e r i e s had r e c e i v e d enough attention to d e s e r v e a
name,
It was v a r i o u s l y called the Braun S e r i e s , the S c h i m p e r - B r a u n
series^ the Lame s e r i e s and the G e r h a r d t s e r i e s .
A. B r a u n [8] , applied
s c a l e s of pine c o n e s ,
to the a r r a n g e m e n t of the
Schimper i s completely unknown.
a l r e a d y been mentioned,
B e r n a r d Lami,
the s e r i e s
Lame''has
but the n a m e h a s been c r e d i t e d to F a t h e r
a c o n t e m p o r a r y of Newton and the d i s c o v e r e r of the
p a r a l l e l o g r a m of f o r c e s .
G e r h a r d t i s probably a m i s - s p e l l i n g of
Girard.
Edouard
Lucas [ 2 0 ] , who dominated
the field
of r e c u r s i v e
s e r i e s during the p e r i o d 1876-1891 , first applied F i b o n a c c i ' s n a m e
to the s e r i e s and it has been known a s Fibonacci s e r i e s since then.
About this t i m e ,
1858, Sam Loyd claimed to have invented the
checkerboard paradox [9].
j o u r n a l in 1868 [ 1 0 ] .
It i s f i r s t found in p r i n t in a G e r m a n
Today i t s e e m s p r o p e r to call it the C a r r o l l
P a r a d o x after Lewis C a r r o l l [11] ( C h a r l e s Dodgson, 1 832-93) who was
quite fond of it.
Before the c e n t u r y ended, a n u m b e r of f a m i l i a r r e l a t i o n s w e r e
found.
Among them:
(V i s the n t h Lucas n u m b e r . )
1876, E , Lucas [12]
u 2 , + u 2 - u 7 .,
n-1
v"
4n
n
Zn-t-1
= vt: - 2
2n
152
FIBONACCI NUMBERS: THEIR HISTORY THROUGH 1900 A p r i l
V
4n+2 = U L + 1
+ 2
u n + p = u n ~ p (u+i) p
u n " p = u n (u-i) p
1886, E. Catalan [13]
[14]
2
n+2 P
Un +,1l - p Un+l+p
. . . - Un+1
" Up
+. = ( - l )
U2 - U
U x = (-l)n"p+1 U .
n
n - p n+p
n-1
1899, E. Landau [15] r e l a t e d the s e r i e s
"i
n=l
( 1'/ U 92n)
to L a m b e r t ' s s e r i e s and
n
to the theta s e r i e s .
~^
A complete list can be found in Vol. 1 of D i c k s o n ' s " H i s t o r y of
the Theory of N u m b e r s . "
This s m a l l h i s t o r y ends a r b i t r a r l y at 1900 for the p r e g m a t i c
r e a s o n that a m e r e listing of twentieth c e n t u r y developments would fill
a m o d e r a t e l y sized volume. It would be i n t e r e s t i n g to see F i b o n a c c i ' s
r e a c t i o n to the application of his rabbit p r o b l e m to such d i v e r s e s u b j e c t s a s m u s i c a l composition [16], p r o c e s s optimization [ 1 7 ] , e l e c t r i c a l network t h e o r y [ 1 8 ] , and genetics [19].
1.
2.
REFERENCES
J. Kepler "De nive s e x a n g u l a " 1611.
B. Concompagni, II Liber Abbaci de Leonardo P i s a n o Pubiicato
da B a l d a s s u r e Boncompagni, Roma MCDDDLVTL
3.
A. G e r a r d .
" L e s O e u v r e s Mathematique de Simon Stevin" Leyde
1634, pp. 169-70.
4.
R. Simpson, "An explanation of an o b s c u r e p a s s a g e in A l b r e c t h
G i r a r d ' s c o m m e n t a r y upon Simon Stevin's w o r k s " P h i l . T r a n s .
Roy. Soc. (London) 48, I, 368-77 (1753).
FIBONACCI NUMBERS; THEIR HISTORY THROUGH 1900
J. P. M. Binet " M e m o i r e sur ['integration des equations
153
lineaires
a u x d i f f e r e n c e s finies d'u,i o r d r e quelconque, a coefficients v a r i a b l e s " Comp. Ren. Accad. Sci. P a r i s 17, 563 (1843).
B. L a m e .
"Note s u r la limite du n o m b r e des divisions dans la
r e c h e r c h e du plus grand commun d i v i s e u r e n t r e deux n o m b r e s
e n t i e r s " Compt. Rend. Acad. Sci. 19, 1867-70 (1944).
E.
Catalan.
"Manual des candidates a Ecole Poly t e c h n i q u e "
tome 1, P a r i s 1857, p. 86.
A. B r a u n .
" V e r g l e i c h e n d e Untersuchung liber die Ordnurng d e r
Schuppen an den Tannenzapfen als Einleitung zur Untersuchung
der
BlattersteHung
uberhaupt.
" Nova
Acta
Acad.
Caes.
Leopoldina 15, 199-401 (1830).
M. G a r d n e r " M a t h e m a t i c s ,
Magic, and M y s t e r y " Dover, New
York, 1956, p . 133.
Zeitschrift fur M a t h e m a t i k und Physik, 13, 162 (1868).
Lewis C a r r o l l " D i v e r s i o n s and D i g r e s s i o n s " Dover, New York,
p. 316-7 (1956).
E. Lucas "Note sur la t r i a n g l e a r i t h m e t i q u e de P a s c a l et sur la
s e r i e de L a m e " Nouvelle C o r r . Math. 2, 74 (1876).
E. Catalan.
Melanges M a t h e m a t i q u e s Vol. 2, Liege 1887, p .
319.
E. Catalan, Mem. Soc. Roy. Sci. Lieges (2), 13, 319-21 (1886).
E. Landau. "Sul la s e r i e des i n v e r s e s des n o m b r e s de F i b o n a c c i
Bull. Soc. Math. F r a n c e 27, 298-300 (1899).
L. Dowling and A. Shaw.
"The Schillinger System of Musical
Composition" C a r l F i s c h e r , New York, 1946.
S, M. Johnson.
"Best Exploration for Maximum is F i b o n a c c i a n "
Rand Corp. R e s e a r c h Memo. RM-1590 (1955).
A. M. M o r g a n - V o y c e .
" L a d d e r - N e t w o r k Analysis using F i b -
onacci N u m b e r s " P r o c . IRE Sept. 1959, pp. 321-2.
F . E. Binet and R. T. Leslie "The Coefficients of inbreeding in
c a s e of Repeated F u l l - S i b - M a t i n g s " J. of Genetics, June I960,
pp. 127-30.
E. Lucas " T h e o r i e des Fonctions Nume'riques Simplement P e r i o d i q u e s " A m . J. Math. 1, 1 8 4 - 2 2 0 ( 1 8 7 8 ) .
ELEMENTARY PROBLEMS AND SOLUTIONS
Edited by A . P . H I L L M A N
University of Santa Clara, Santa Clara, California
Send all c o m m u n i c a t i o n s r e g a r d i n g E l e m e n t a r y P r o b l e m s and
Solutions to P r o f e s s o r
A. P . Hillman,
Mathematics
U n i v e r s i t y of Santa C l a r a , Santa C l a r a , California.
Department,
We w e l c o m e any
p r o b l e m s believed to be new in the a r e a of r e c u r r e n t sequences a s
well a s new a p p r o a c h e s to existing p r o b l e m s .
The p r o p o s e r should
s u b m i t his p r o b l e m with solution in legible form, p r e f e r a b l y typed in
double spacing, with n a m e ( s )
and a d d r e s s of the p r o p o s e r
clearly
indicated.
Solutions to p r o b l e m s listed below should be submitted within
two months of publication.
B-38
Proposed
by Roseanna
Torretto,
University
of Santa Clara, Santa Clara,
C h a r a c t e r i z e s i m p l y all the sequences
n+2 ~
B-39
Proposed
by John Allen
Fuchs,
n+1
University
c
California
satisfying
n
of Santa Clara,
Santa Clara,
California
Let F , = F . = 1 and F ^ = F ,. + F for n > 1 .
1
2
n+2
n+1
n
P r o v e that
F ^ < 2 n for
n+2
B-40
Proposed
If H
n
by Charles
R. Wal/, Texas Christian
n > 3 .
University,
Fort Worth,
Texas
is the n - t h t e r m of the g e n e r a l i z e d F i b o n a c c i s e q u e n c e ,
°
i.e.,
154
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
R
1
= pr , H
2
155
= pr + q , H ± = H ± 1 + H
forn>l,
^
n+2
n+1
n
show that
2 kH. = x(n + 1)H
q .
;
J - - H , , + 2p + H
, 2-,
.*
n+2
n+4
^
k=l
B-41
Proposed
by David L. Silverman,
Beverly
Hills,
California
Do t h e r e e x i s t four d i s t i n c t positive F i b o n a c c i n u m b e r s in a r i t h metic progression?
B-42
Proposed
by S.L. Basin,
Sylvania
E x p r e s s the (n + l ) - s t
of F .
n
B-43
Proposed
Electronic
Systems, Mountain
Fibonacci n u m b e r
View,
F ,,
as a function
Also solve the s a m e p r o b l e m for Lucas n u m b e r s .
by Charles
R. Wo//, Texas Christian
Fort Worf/i, Texas
University,
(a) Let x n > 0 and define a sequence x,
k > 0S w h e r e f(x) = y l + x.
by x,
Find the l i m i t of x,
(b) Solve the s a m e p r o b l e m for
f(x) =
v 1 + 2x
(c) Solve the s a m e p r o b l e m for f(x) =
V2 + 3x
(d)
California
, = f(x, ) for
as k —• ©o.
Generalize.
SOLUTIONS
FIBONACCI AND PASCAL AGAIN
B-16
Proposed
by Mar/or/e Bicknell,
Terry Brennan,
Show that if
Lockheed
San Jose State College,
Missiles
San Jose,
and Space Co., Sunnyvale,
California,
California
and
156
April
ELEMENTARY PROBLEMS AND SOLUTIONS
then
2F
n-1
Rx"
F
,F
n-1 n
,F
n-1 n
F
- F
n+1
2F
F
n-1
F
n
F
4-1
n n+1
nFn+l
n+1
( T h e r e a r e s o m e m i s p r i n t s in the originaL s t a t e m e n t . )
Solution hy\L. Carlitz, Duke University,
Durham, N.C.
Put
R
a m a t r i x of o r d e r
k + 1; for example
0
R
l
(r, s = 0, 1, . . . , k) ,
(.'.)
k=
1
R
=
- 1
"0
0
1"]
0
1
1
_1
2
1J
2 "
1_
It is e a s i l y verified that
F
(1)
R;
.
n-1
F
n
( n = 1, 2,
...)
n+1
Indeed this is obviously t r u e for
n = 1.
A s s u m i n g that the formula
holds for n, we have
n+1
R"
n-1
n+1
n+1
1
1
L
n+1
In the next place we notice that the t r a n s f o r m a t i o n
n+2
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
x' =
!
157
y
y1 = x + y
induces the t r a n s f o r m a t i o n s
,2
:'
=
T2:
x'3
.1
=x
+ 2xy + y
=
,2 ,
x 1 y1 =
, ,2
x'y1 =
y'
and so on.
xy + y 2
^ x'y« =
y
T
2
y
y
Z
3
+ y
2
2 , 3
x y + 2xy + y
'
= x
+ 3x y + 3xy
A l s o it is evident from (1) that T,
Tn.
1
4
xy
+ y
is given by
\ x ^ = F ,x.+ F y7
/
n-1
n
) y ( n ) = F n~
X +'
F
n+ly
We t h e r e f o r e get
(x x ( n ) ) 2
'
= F 2 , x 2 + 2 F . F xyJ + F 2Jy 2
n-1
n-1 n
n
2
„n ) x (n) (a) = F
T-:
j x ' Vs
= Fn-1
n_lFnx
( n )2 2
(wyM)
)
'
+
(F 2
+
F^F^Jxy +
2
2
,= FF 22x X2 2+ +2 F— F ,. xy
- • + ~F~
,, yn
n n+1 J
n+1 J
A/so solved by the proposers.
LAMBDA FUNCTION OF A MATRIX
B-24
Proposed by Brother U. Alfred, St. Mary's College, California
It is evident that the d e t e r m i n a n t
F^y 2
158
E L E M E N T A R Y P R O B L E M S AND SOLUTIONS
h a s a v a l u e of z e r o .
n+1.
n+2
n+1
n+2
n+3
n+2
n+3
'n+4
P r o v e t h a t if t h e s a m e q u a n t i t y
k is added to
t h e v a l u e b e c o m e s (-1) n - 1 k.
e a c h e l e m e n t of t h e a b o v e d e t e r m i n a n t ,
Solution
April
by Raymond Whitney, Pennsylvania
State University,
Hazelton
Campus
U s i n g& t h e b a s i c F i b o n a c c i r e c u r s i o n f o r m u l a F
e l e m e n t a r y row and c o l u m n t r a n s f o r m a t i o n s
,, + F a n d
in=F
n+2
n+1
n
we m a y r e d u c e the d e -
t e r m i n a n t to:
n+1
F
n+1
F
n+2
-1
-1
0
which is
(-1)
k
= k(F F
n n+2 • F n + 1 >
•
1
by a b a s i c i d e n t i t y .
Also solved by Mor/or/e Bicknell,
who p o i n t e d out the
San Jose State College, San Jose,
r e l a t i o n to
"Fibonacci
California
Matrices
and
Lambda
F u n c t i o n s , " b y M, B i c k n e l l a n d V. E . H o g g a t t , J r . , t h i s Q u a r t e r l y ,
V o l . 1, N o . 2; R. M . G r a s s l ,
F.D. Parker, State University
U n i v e r s i t y of S a n t a C l a r a ,
California;
of New York, Buffalo, N.W., R.N. Vawter, St. Mary's
College,
California; H.L. Walton, Yorktown H.S., Arlington, Virginia; and the proposer.
EXPONENTIALS OF FIBONACCI NUMBERS
B-25
Proposed by Brother U. Alfred, St. Mary's College,
Find an e x p r e s s i o n for the
T n = 1, T 1 = a,
T 9 = a,
...
In-I
2n
2n-2
California
general term(s)
of t h e
where
and
"2n+l
,T7 T9
.
Zn 2 n - 1
.
sequence
1964
ELEMENTARY PROBLEMS AND SOLUTIONS
159
Solution by Vassili Daiev, Sea Cliff, LA., N.Y.
The f i r s t few t e r m s a r e
F
F
0
It is e a s y1 to see that
n
k = F,(n+3)/2
IO \ /o ^
F
2 , 1
T
n
is odd.
Also solved by J.AM.
= a
F
F
3
F
2
4
w h e r e k = F , '/ o x if n is even and
(n/2)
Hunter, Toronto,
Ontario,
Canada
who suggested the c o n s i d e r a t i o n of log T ;
Ralph Vawter, St. M a r y ' s College, California, and the p r o p o s e r .
E d i t o r i a l Comment: The p r o b l e m can be solved by showing that
log T L . = log T
+ log T ,
&
mL4
° m+2
° m
MAXIMIZING A DETERMINANT
B-28
Proposed by Brother U. Alfred, St. Mary's College, California
Using the nine F i b o n a c c i n u m b e r s F ? to F , ~ (1, 2, 3, 5, 8, 13,
21,
34, 55),
d e t e r m i n e a t h i r d - o r d e r d e t e r m i n a n t having e a c h of
t h e s e n u m b e r s a s e l e m e n t s so that the value of the d e t e r m i n a n t is a
maximum0
Solution by Mar/one Bicknell,
San Jose State College,
California
By c o n s i d e r i n g combinations of F i b o n a c c i n u m b e r s which give
m i n i m u m and m a x i m u m values to s u m s of the form
abc + def + ghi,
the following d e t e r m i n a n t s e e m s to have the m a x i m u m value obtainable with the nine F i b o n a c c i n u m b e r s given:
F .10 F4 *7
F
F10F9F3 + F?F6F5 + F4F3F2 - ( F ^ F g + F ^ F ,
F6
F9 F3
F2
F 5 F8
+F
39796 "
38300.
Also solved
by the
proposer.
1496
8 F 4 F 6>
160
B-29
E L E M E N T A R Y P R O B L E M S AND SOLUTIONS
Proposed by A.P. Boblett,
U.S. Naval Ordnance Laboratory,
April
Corona, California
Define a g e n e r a l F i b o n a c c i sequence such that
F , = a;
1
F 0 = b;
2
F
F
n
n
=F
n -02
+ F
.,
n-1
= F , - - F ,. ,
n+2
n+1
Also define a c h a r a c t e r i s t i c n u m b e r ,
n > 3
n < 0
"~
C> for this sequence, w h e r e
C = (a + b)(a - b) + a b .
Prove:
F , . F , - F 2 = ( - l ) n C , for a l l n .
n+1 n - 1
n
Solution by P.O. Parker, State University
From
of New York, Buffalo,
N.Y.
F(n) = F(n - 1) + F(n - 2), F ( l ) = a, F(2) = b,
we get
_,
b - a r n . b - as n
F(n)x =
-j s + — — j r ,
1 +s
1 +r
where
r and s a r e solutions of the q u a d r a t i c x
2
- x - 1 = 0.
Using
the fact that r + s = - r s = 1, d i r e c t calculation yields
F(n + l ) F ( n - 1) - F 2 ( n ) = [(a - b)(a + b) + ab ] ( - l ) n .
The well known r e s u l t
F(n + l ) F ( n - 1) - F 2 ( n ) = ( - l ) n is the
s p e c i a l c a s e in which a = b = 1.
Also solved by Mor/or/e Bicknell,
San Jose State College, San Jose,
Donna J. Seaman, Sylvania Co.; R.N. Vawter, St. Mary's College,
the proposer, J.A.H. Hunter, of Toronto,
Ontario.
California;
California; and
