Erdős Problem #141

Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?

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Green and Tao [GrTa08] have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erdős called this conjecture 'completely hopeless at present'.

The existence of such progressions for small $k$ has been verified for $k\leq 10$, see the Wikipedia page. It is open, even for $k=3$, whether there are infinitely many such progressions.

See also [219].

This is discussed in problem A6 of Guy's collection [Gu04].

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This page was last edited 28 September 2025.

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Related OEIS sequences: A006560

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Additional thanks to: Prakrish Acharya

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