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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

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    $\begingroup$ Have you heard about quadratic residues? $\endgroup$ – Seewoo Lee Mar 5 '17 at 11:22
  • $\begingroup$ Odd prime factors, maybe? $\endgroup$ – Oscar Lanzi Mar 5 '17 at 11:29
  • $\begingroup$ If you seek yo prove it directly, that is a proof that constructs the result for all of them. If it is true for infinite $n$ then you require induction or contradiction. For contradiction it would be like assuming there exists an $n$ such that the premis isnt true or factors not congruent to 1 or 5. Then deducing a contradiction would show that they must at least be congruent to 1 or 5. Just to explain that if it helps. Induction would also work. $\endgroup$ – marshal craft Mar 5 '17 at 12:10
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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 $p=2$.

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  • $\begingroup$ In case you have never seen it, the fact that $-1$ being a quadratic residue modulo $p$ implies $p\equiv1\pmod4$ is explained e.g. here. Probably also earlier, that is one of the most often used tricks in elementary number theory. $\endgroup$ – Jyrki Lahtonen Mar 5 '17 at 11:40
  • $\begingroup$ correct to match one part of the answer $\endgroup$ – mmath Mar 5 '17 at 11:40
  • $\begingroup$ @mmath: Sorry, what do you mean by part of the answer? I don't think there is anything missing :-) $\endgroup$ – Jyrki Lahtonen Mar 5 '17 at 11:42
  • $\begingroup$ ok, my apologise . everything is there $\endgroup$ – mmath Mar 5 '17 at 11:45
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    $\begingroup$ Correct, @Student. Sorry about that. I edited the comment (one of the diamond superpowers is the ability to edit old comments). $\endgroup$ – Jyrki Lahtonen Mar 5 '17 at 12:29

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