The Green-Tao theorem asserts that for any $k$, there exists an arithmetic progression of length $k$ consisting only of primes. What I was wondering, is whether it has been proven whether there are infinitely many such progressions for any $k$, or for some $k$, the number of prime-only progressions of length $k$ is finite.
2026-03-28 01:06:36.1774659996
Extension of the Green-Tao theorem
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It is trivial to see that we will get infinitely many. Just consider prime AP's of length nk where you choose n. Then u get n such prime AP's of length k trivially.