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I've seen various references to Dirichlet's theorem on arithmetic progressions claiming that where $a, b$ coprime, $ax + b$ not only generates infinitely many primes, but also infinitely many semiprimes (or 2-almost primes), infinitely many 3-almost primes, etc., and indeed, creates infinitely many $k-$almost primes for any arbitrarily large $k$. None of these sources in which I've seen this claim I would consider to be definitive. So my question is: is this actually true or just fanciful conjecture?

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That would be the same as Dickson's Conjecture because assuming $a$, $b$, $n$, $n_2$, $n_3$... are pairwise coprime, there is some integer $x$ such that $ax+b$ divides $n_z$ by modular arithmetic. To show that there are infinitely 2, 3, 4 etc. almost-primes... would mean there are infinitely many primes of the form ($ax+b$)/$n$ (assuming $n_z$ is prime) and ($ax+b$)/($n$$n_2$$n_3$...) an implication of Dickson's Conjecture since the forms are linear.

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