Polignac's Conjecture

Prove that for every positive integer m, there are infinitely many consecutive-prime indices n such that p_{n+1}-p_n=2m.

Background

The paper describes Polignac's conjecture as asserting the existence of infinitely many consecutive prime gaps of every positive even size. It notes that the case m=1 is the twin prime conjecture and that Zhang's bounded-gap theorem establishes only a weakening for some value of m.

A family of generalized Ramsey numbers is then constructed whose existence for all positive integers t and m is claimed to be equivalent to Polignac's conjecture. The conjecture itself remains unproved in the paper.

References

For every m\in N there are infinitely many values n\in N such that p_{n+1}-p_n = 2m.

— Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems  (2502.04311 - Christopherson, 6 Feb 2025) in Section 6, immediately before Theorem 6.5, Conjecture (Polignac)

As is well-known, Zhang's remarkable proof verifies a particular case of Polignac's conjecture . For every $m\in N$ there are infinitely many values $n\in N$ such that $p_{n+1}-p_n = 2m.

— Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems  (2502.04311 - Christopherson, 6 Feb 2025) in Section 6, subsection beginning “Polignac”