91Ó°ÊÓ

(a) Let \(p\) be an odd prime. If \(p \mid a^{2}+b^{2}\), where \(\operatorname{gcd}(a, b)=1\), prove that the prime \(p \equiv 1(\bmod 4) .\) [Hint: Raise the congruence \(a^{2} \equiv-b^{2}(\bmod p)\) to the power \((p-1) / 2\) and apply Fermat's theorem to conclude that \(\left.(-1)^{(p-1) / 2}=1 .\right]\) (b) Use part (a) to show that any positive divisor of a sum of two relatively prime squares is itself a sum of two squares.

Prove that every odd integer is the sum of four squares, two of which are consecutive. [Hint: For \(n>0,4 n+1\) is a sum of three squares, only one being odd; notice that \(4 n+1=(2 a)^{2}+(2 b)^{2}+(2 c+1)^{2} \quad\) gives \(\quad 2 n+1=(a+b)^{2}+(a-b)^{2}+c^{2}+\) \(\left.(c+1)^{2} \cdot\right]\)