Math Help Forum - View Single Post

Bruno J. · #4 June 17th, 2009, 02:50 PM

Suppose $p=a^2+b^2=c^2+d^2$ .
Then

$p=(a+bi)(a-bi)=(c+di)(c-di)$

and each of $a \pm bi, c \pm di$ are primes in $\mathbb{Z}[i]$ because they have prime norms.
But $\mathbb{Z}[i]$ is a unique factorization domain hence the primes are the same up to unit factors and rearrangement (i.e. : $a+bi \in \{\pm(c+di), \pm i (c+di)\}$ . This yields that $\{a^2,b^2\}=\{c^2,d^2\}$ and we are done.