The arithmetic progression $a_N=(p-1)N+1$ contains infinitely many primes $q$ by Dirichlet. I have searched this part in wiki, but I din't get any relevant proof. Can any one prove it how $a_N$ contains infinitely many primes and what is Dirichlet proof? please explain.


marked as duplicate by davidlowryduda Oct 20 '17 at 5:26

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    $\begingroup$ There are many places it is done, for example Apostol's book on Analytic Number Theory. For the case $cn+1$, which is the one you are interested in, there is also a proof that does not use Analytic Number Theory. But it is not easy. Neither proof can, I think, be usefully summarized in an MSE answer. $\endgroup$ – André Nicolas Feb 15 '14 at 4:11
  • $\begingroup$ @AndréNicolas! I don't have the book. Any how, if you can summarize this part, I will be happy. Please do it. $\endgroup$ – mooorthyannaya Feb 15 '14 at 6:05
  • $\begingroup$ Searching will show that there are many complete (and lengthy!) proofs on the Internet. Here is the first one I bumped into. I do not know how clear it is, mentioned Apostol because I am familiar with it and it is good. $\endgroup$ – André Nicolas Feb 15 '14 at 6:19
  • $\begingroup$ @AndréNicolas! Thanks a lot. but nothing going in my mind. $\endgroup$ – mooorthyannaya Feb 15 '14 at 6:59
  • $\begingroup$ You are welcome. The result is difficult, unless you have a fairly considerable amount of background knowledge. $\endgroup$ – André Nicolas Feb 15 '14 at 7:03

He used the L-function. You can read about it on this page - https://en.m.wikipedia.org/wiki/Dirichlet_L-function

It says that he introduced them to prove his theorem.


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