prime factors of $ n^{2}+4$ is congruent to 1 or 5 (mod 8)

Question

prove that the prime factors of $n^{2}+4$ , $ n \in \mathbb{N}$ are congruent to $ 1 \ or \ 5 \ (mod \ 8)$. $$$$ I can that the statement holds for n=1,2,3 , so can i use induction principle ? If there is any other way to solve this? Thanks

Answer 1

If $p$ is a prime factor of $n^2+4$ then $n^2\equiv-4\pmod p$, and $-4$ is a quadratic residue modulo $p$. Because $4$ is a square this is equivalent to $-1$ being a quadratic residue modulo $p$. It is well known that this is so if and only if $p\equiv1\pmod4$ (with the obvious exception $p=2$).

Therefore $p$ is congruent to either $1$ or $5$ modulo $8$, or $p=2$.