Twin Prime Conjecture

Prove that there are infinitely many pairs of consecutive primes whose difference is 2; equivalently, establish that infinitely many indices n satisfy p_{n+1}-p_n=2.

Background

The paper reformulates the twin prime problem in terms of the existence of a family of generalized Ramsey numbers constructed from metrical colorings of complete graphs. It states that the existence of the relevant Ramsey numbers for every starting index m is equivalent to the twin prime conjecture.

The conjecture remains unresolved in the paper and is presented as an example of a classically non-Ramseyian problem that can be expressed in Ramsey-theoretic language.

References

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

— Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems  (2502.04311 - Christopherson, 6 Feb 2025) in Section 6, subsection following Corollary 6.2, Conjecture (Twin Prime Conjecture)

Let's take a look at another example of this phenomenon in the context of a conjecture that is, as of yet, unproven. There are infinitely many values $n\in N$ such that $p_{n+1} - p_n = 2.

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

There are infinitely many values $n\in N$ such that $p_{n+1} - p_n = 2$.

— Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems  (2502.04311 - Christopherson, 6 Feb 2025) in Section 6, immediately before Theorem 6.4 (the theorem labeled 'twin prime')

If there are infinitely many twin primes (an open question), then this program is circle-free.

— Did Turing prove the undecidability of the halting problem?  (2407.00680 - Hamkins et al., 2024) in Subsection "Circle-freeness", Section "The circle-free problem"

DHL[2,2] implies the twin prime conjecture via the admissible 2-tuple $(0,2)$, a conjecture that remains open.

— Generative Modeling for Mathematical Discovery  (2503.11061 - Ellenberg et al., 14 Mar 2025) in Section 3, subsection “Narrow admissible tuples”