MathDB

Let

p

be a prime number of the form

4k+1

. Suppose that

r

is a quadratic residue of

p

and that

s

is a quadratic nonresidue of

p

. Show that

p=a^{2}+b^{2}

, where

a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).

Here,

\left( \frac{k}{p}\right)

denotes the Legendre Symbol.

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