Weinstein 2

A L E T T E R TO THE EDITOR

88

Feb.

3) If one s u m s the coefficients in the table without f i r s t multiplying by
p o w e r s of five one obtains k - t h power r e s i d u e s of two.
Some of this has undoubtedly been o b s e r v e d before and even
probably proved but we have no idea how m u c h .
We have enjoyed playing around with t h e s e concepts and actually
s u s p e c t much m o r e than we have indicated h e r e .

If anyone is i n t e r -

e s t e d in p u r s u i n g this further, we shall be glad to h e a r from h i m .
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LETTER TO THE EDITOR
Lenard Weinstein
Massachusetts Institute of Technology

Conjecture 2 . , m a d e by M r . Thoro, on page 186 of the October
i s s u e , follows i m m e d i a t e l y from the following t h e o r e m found on page
126 of W. J. Leveque, Topics in Number Theory, Vol. I:
Definition
A r e p r e s e n t a t i o n of a positive i n t e g e r n as a sum of two s q u a r e s ,
2
2
say n = x + y is t e r m e d p r o p e r if (x, y) = 1.
Theorem
If p is a p r i m e of the form

4k + 3 and p |n, then n h a s no

proper representation.
Since

F 0 ,. = F + F ,_ , and ( F , F , . ) = 1, F - ,. always
J
2n+l
n
n+1
n n+1
2n+l
has a p r o p e r r e p r e s e n t a t i o n .
T h e r e f o r e , by the above t h e o r e m , no
pc r i m e of the form 4k + 3 can divide F - ,, .
2n+l
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