We have to integers $a,b$. I need to show that if $a$ and $b$ are coprimes then the set of prime numbers of kind $ak+b$ is infinite.
How could I show it ? I know how to do that for $4k+3$ or $4k+1$, but I have no idea how to get the general answer.
2026-04-02 10:18:21.1775125101
Prove that there are an infinity of prime $ak+b$, $a$ and $b$ coprimes
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As the comments have mentioned, this is Dirichlet's theorem on arithmetic progressions and requires some fairly involved analytic number theory. There is probably not a simple proof.
You can find an English translation of Dirichlet's original paper online, but I haven't checked the translation for accuracy, relevance, or legibility (because it is beyond me!)