I have these problems assigned for homework, I was able to get through the rest of the homework fine but I'm stuck on these two. Prove that
- If $n\equiv 6\pmod9$, then $n$ cannot be a sum of two integer squares.
- If $n\in\mathbb{N}$ is not the sum of two integer squares, then it is not the sum of two rational squares.
I've been trying to use these theorems from class but I haven't gotten far:
Theorem 8.2: For $p$ an odd prime, there exists $a\in\mathbb{N}$ so $a^2\equiv -1\pmod{p}$.
Theorem 8.3: For $p$ an odd prime, there exist $a,b\in\mathbb{N}$ so $p=a^2+b^2\iff p\equiv 1\pmod4$.
Theorem 8.4: $n\in\mathbb{N}$ is a sum of two squares in $\mathbb{Z}\iff$ every prime divisor $p$ of $n$ with $p\equiv 3\pmod4$ occurs an even number of times in the prime factorization of $n$.