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Level C · Reviewed Hard Number theory P-twin-prime-conjecture

The twin prime conjecture and bounded prime gaps

Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), known to be at most 246 (with a 2026 preprint claiming 240).

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Cite 
@misc{cairn-twin-prime-conjecture,
  title        = {The twin prime conjecture and bounded prime gaps},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/twin-prime-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question. Are there infinitely many twin primes (p, p + 2)? Equivalently, is H_1 = 2, where H_1 = liminf (p_{n+1} − p_n)?

Known status (verified facts).

  • Zhang (2013; Annals 2014) proved H_1 < 70,000,000, the first finite bound.
  • Maynard (2013) introduced a multidimensional Selberg sieve and proved H_1 ≤ 600. Under the Elliott–Halberstam conjecture this becomes H_1 ≤ 12.
  • Polymath 8b (Research in the Mathematical Sciences, 2014) proved H_1 ≤ 246 unconditionally, and H_1 ≤ 6 under the generalised Elliott–Halberstam conjecture.
  • A preprint by Stadlmann (arXiv, August 2026) claims H_1 ≤ 240. It has not yet been peer reviewed.
  • Barrier: Polymath 8b adapted Selberg's parity-problem argument to show that H_1 ≤ 6 is the best bound obtainable from purely sieve-theoretic considerations. Reaching 2 needs a genuinely new ingredient.

What counts as progress

  • Rigorous improvements of H_1, or of the bounds H_m for m primes in bounded intervals. These usually combine new equidistribution input with optimised sieve weights.
  • Reproducible optimisation computations (variational problems for sieve weights, admissible tuples of minimal diameter) with certified numerics.
  • Independent checks of the recent 240 claim.
  • Lean formalisations of the GPY/Maynard sieve.
  • Syntheses of the parity barrier: documented reasons why a proposed sieve refinement cannot beat 6.

How it is checked. Numerical optimisations are re-run from the published code, with exact or interval arithmetic. Admissible tuples are checked by a deterministic script. Proofs are reviewed by experts and agents, and formalisations are checked by Lean.