Is it necessary for a number of the form $4k^2+1$ to have at least one prime factor of the form $4n+1$?
I was trying to prove that there are infinitely many primes of the form $4n+1$, but to prove it, I need to prove the above statement true.
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Vibhav Aggarwal
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Even more : EVERY prime factor of such a number must be of the form $4n+1$ – Peter Jan 13 '19 at 15:53
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@Peter I would really appreciate if you provide a proof for that. – Vibhav Aggarwal Jan 13 '19 at 15:55
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3Considering quadratic residues, the proof is trivial. Every prime factor $p$ must be odd, and since the number is of the form $n^2+1$ , $-1$ is a so-called quadratic residue mod $p$ , whenever $p$ divides $4k^2+1$ , we can conclude immediately that $p$ must be congruent to $1$ modulo $4$. – Peter Jan 13 '19 at 16:05
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Cf. here – J. W. Tanner Jan 13 '19 at 16:11
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Also there are infinitely many primes of the form $4n+1$ trivially by Dirichlet's theorem on arithmetic progressions. – Tejas Rao Jan 16 '19 at 17:12