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I was thinking about primes earlier and I thought of a hypothesis that I have been unable to prove. I was wondering whether it was a known theorem and whether anyone knows a proof or can prove (or disprove) it.

here it is:

there are an infinite number of primes that satisfy the equation:

$p \equiv a \bmod n$

for all A and N where A and N are relatively prime.

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    $\begingroup$ That's Dirichlet's theorem about primes in arithmetic progressions. It's not easy to prove it. $\endgroup$ – Daniel Fischer May 10 '14 at 20:27
  • $\begingroup$ oh sweet, thank you. do you say that to mean it has not been proven or that the proof is too advanced for this post? $\endgroup$ – maxG795 May 10 '14 at 20:30
  • $\begingroup$ Dirichlet proved it. I suspect the proof is beyond what fits in a reasonable answer on the site, but I may be wrong in that. $\endgroup$ – Daniel Fischer May 10 '14 at 20:34
  • $\begingroup$ @maxG795: Please check the correction I applied in your equation, to make sure it's correct. $\endgroup$ – barak manos May 10 '14 at 20:55
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This is Dirichlet's theorem on arithmetic progressions of primes. A brief sketch of the proof is given in the article, along with a reference to Jürgen Neukirch's Algebraic number theory $(1999)$.

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