Proving that $7n+3$ is prime infinitely often?

Question

The Question is: "Prove that there are infinite values of natural $n$ for which $7n+3$ is prime.

Edit: I have realised that this is a direct consequence of Dirichlet's Theorem, but the proof of this theorem is beyond me at the moment.

Is there any simple/elementary way to prove this statement?

I think you mean infinitely many values of natural $n$. Do you mean $5n+3$ or $7n+3$? Either follows from Dirichlet's theorem — J. W. Tanner, Nov 11 '20 at 18:10
This is pretty much the same as saying "Prove that $n$ is prime for infinitely many $n$s". This is obviously true. — talbi, Nov 11 '20 at 18:11
@J.W.Tanner Thank you for the reference. I did not know of this theorem.. — Aditya Sharma, Nov 11 '20 at 18:13
@talbi I do not see how it is obvious nor do I understand the way you have reframed the problem could you please elaborate? — Aditya Sharma, Nov 11 '20 at 18:15
Euclid's Theorm proved that there exists infinitely many prime numbers. So, there exists infinitely many $n$s, for which $n$ is prime. Of course, for $\alpha n + \beta$ this is not as simple, and requires Dirichlet's Theorem. — talbi, Nov 11 '20 at 18:21
@AdityaSharma: You're welcome. You might be interested in this article — J. W. Tanner, Nov 11 '20 at 18:32

lhf · Accepted Answer · 2020-11-11T18:54:34.710

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An elementary proof along the lines of Euclid's proof exists for the arithmetic progression $a \bmod n$ iff $a^2 \equiv 1 \bmod n$. Since $3^2 \not\equiv 1 \bmod 7$, there is no such proof.

Here are some references:

Primes in Certain Arithmetic Progressions by Murty and Thain.
How I discovered Euclidean proofs by Murty.
Euclidean proofs of Dirichlet's theorem by Keith Conrad.

edited Nov 11 '20 at 18:54

answered Nov 11 '20 at 18:48

lhf

216,483

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Adapted from https://math.stackexchange.com/a/3346456/589 – lhf Nov 11 '20 at 18:49

Proving that $7n+3$ is prime infinitely often?

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