## Distribution of square residuals mod prime

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• If p is a positive prime equal to 1 mod 4, then each square residual mod p is the difference of non-zero squares mod p in (p-5)/4 ways, and each non-square
Message 1 of 1 , Jul 17, 2009
If p is a positive prime equal to 1 mod 4,

then each square residual mod p
is the difference of non-zero squares mod p
in (p-5)/4 ways,

and

each non-square residual mod p,
is the difference of non-zero squares mod p
in (p-1)/4 ways.

This is consistent with
Number of nonzero square residual pairs with non square residual difference
+ number of nonzero square residual pairs with square residual difference
+ number of nonzero square residual pairs with zero difference
= ((p-1)/2) **2

If p = 3 mod 4,
then
each nonzero element mod p
is the difference of non-zero squares mod p
in (p-3)/4 ways.

This is consistent with

Number of nonzero square residual pairs with non square residual difference
+ number of nonzero square residual pairs with square residual difference
+ number of nonzero square residual pairs with zero difference
= ((p-1)/2) **2

Kermit Rose
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