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This is the full question

Let $P$ be an odd prime. Prove that if there is an integer $x$ such that,

$$ p\mid x^2+1 \text{ then } p \equiv 1 \pmod 4 $$

$$ p\mid x^2-2 \text{ then } p \equiv 1 \text{ or } 7\pmod 8 $$

$$ p\mid x^2+2 \text{ then } p \equiv 1 \text{ or } 3\pmod 8 $$

$$ p\mid x^4+1 \text{ then } p \equiv 1 \pmod 8 $$

Show that there are infinitely many primes of each of the forms $8n+1,8n+3,8n+5,8n+7$

I was able to show all the above four relations, but i don't understand how these imply that there are infintely many such primes.

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Sam
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  • The first one should me "if $p\mid x^2+1$ then $p\equiv 1\pmod 4$", the second one should be "if $p\mid x^2-2$ then $p\equiv 1$ or $7\pmod 8$". – Wojowu Sep 16 '19 at 09:48
  • yes you are right @Wojowu, typo corrected – Sam Sep 16 '19 at 09:50
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    A somewhat surprising fact is that an elementary proof (not invoking complex analysis techniques as in Dirichlet) is available for the infinitude of primes $p$ with $p\equiv a\pmod n$ if and only if $a^2\equiv1\pmod n$. With $n=8$ this holds for all odd $a$ covering this question nicely. I collected some links when preparing this question. – Jyrki Lahtonen Sep 16 '19 at 11:31
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    The key here is that if there are a finite number of primes of any type, you can exclude them from being factors by choosing $x$ carefully. If you also choose an expression which is guaranteed to have at least one prime factor of your target type, you will have found a prime not on the original list. – Mark Bennet Sep 16 '19 at 12:24

2 Answers2

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Say you have $p|x^2+2$ implies $p\in\{1,3\}\bmod 8$. Now suppose there are only finitely many primes of the form $8n+3$. Let $\Pi$ be the product of these primes and consider the combination

$M=\Pi^2+2$

None of the primes used to make $\Pi$ can be a factor of $M$ and the actual prime factors of $M$ cannot be all of the form $8n+1$ because $M\equiv 3\bmod 4$. We are forced to allow more $8n+3$ prime factors, thus the proposed finite set of such primes could not have contained all of them.

Use your other expressions to render similar proofs for the other cases.

Oscar Lanzi
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HINT.-All primes are of the form $4n+1$ or $4n+3$ or also of the form $8n+i$ where $i=1,3,5,7$. From your first four propositions you have proved you can deduce easily your four arithmetic progressions.Now you have to apply Dirichlet theorem to each of them.

Piquito
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