Question 17
(FRA 3) Prove that for any positive integers $x,y,z$ with $xy-z^2=1$ one can find nonnegative integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.
Set $z=(2q)!$ to deduce that for every prime $p=4q+1$, the number $p$ can be represented as the sum of the squares of two integers.
Set $z=(2q)!$ to deduce that for every prime $p=4q+1$, the number $p$ can be represented as the sum of the squares of two integers.