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Finite prime gaps from weighted Selberg sieves and pair penalties

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Annanay Kapila

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

Let Hm=lim inf⁡n→∞(pn+m−pn)H_m=\liminf_{n\to\infty}(p_{n+m}-p_n). We develop a pair-penalty refinement of a multidimensional Selberg sieve, together with explicit admissible tuples and interval-arithmetic evaluations. A mixed prime–divisor asymptotic is derived from distribution on triply densely divisible moduli. Combining it with the numerical certificates gives H2≤19,844H_2\le19{,}844, H3≤1,357,780H_3\le1{,}357{,}780, H4≤82,234,788H_4\le82{,}234{,}788, and H5≤4,830,355,416H_5\le4{,}830{,}355{,}416. The proof combines truncated marginal estimates, support-loss bounds for signed trial functions, a trace-preserving auxiliary sieve for prime pairs, and exact modular certificates. Exact finite data, interval enclosures, and reproducibility code accompany the manuscript.

Author comments: Updated manuscript dated 6 November 2026, with a pair-penalty refinement and improved bounds for H_2 through H_5.

Provenance statement

The text is primarily AI-generated. Some parts of this contribution, but not all, have been checked and understood by a human. The manuscript is accompanied by interval-arithmetic evaluations, exact finite certificates and reproducibility code.

References

  1. D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes , Research in the Mathematical Sciences 1 (2014), article 12. Version 4, 22 December 2014. Theorem 2.3, Theorem 3.6(i), Lemma 4.1 and §5.3. https://arxiv.org/abs/1407.4897v4.link
  2. OpenAI, Improved short gaps between primes , preprint, 30 August 2026. Definition 2.1, equation (2.5), Proposition 2.14, Proposition 2.18, Corollary 2.19 and Appendix A.4. https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf.link
  3. J. Stadlmann, On primes in arithmetic progressions and bounded gaps between many primes , version 3, 22 February 2025. Proposition 1 and §2.1.1. https://arxiv.org/abs/2309.00425v3.link
  4. G. Sarajian and A. Weingartner, An extension of smooth numbers: multiple dense divisibility , version 2, 24 September 2025. Theorem 2 and Lemma 2. https://arxiv.org/abs/2504.02699v2.link
  5. The MPFR team, GNU MPFR 4.2.2 manual , sections on rounding modes, functions, and IEEE arithmetic. https://www.mpfr.org/mpfr-current/mpfr.html.link

Version history

  1. v1 · 17 pagesSubmitted by Annanay KapilaInitial depositSupersededOct 20, 2026
  2. v2 · 35 pagesSubmitted by Annanay KapilaUpdated manuscript dated 6 November 2026, with a pair-penalty refinement and improved bounds for H_2 through H_5.CurrentNov 10, 2026