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Improved bounds on HmH_m for 2≤m≤52\le m\le 5

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

Contributed by Annanay Kapila ↗

Version 1 / 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), where pnp_n denotes the nnth prime. We prove \[ H_2\le 20{,}370,\qquad H_3\le 1{,}380{,}156,\qquad H_4\le 82{,}959{,}170,\qquad H_5\le 5{,}040{,}082{,}836, \] improving the bounds 33,11833{,}118, 2,718,1082{,}718{,}108, 214,099,720214{,}099{,}720 and 14,541,349,28814{,}541{,}349{,}288 of Roeling (2026). We use the multidimensional Selberg sieve of Maynard and Tao in the form developed by Polymath. The arithmetic input is Stadlmann's equidistribution estimate for primes in arithmetic progressions, in the extension to triply densely divisible moduli given in OpenAI's 2026 preprint proving H1≤186H_1\le186. Together with a criterion from the same preprint for the triple dense divisibility of least common multiples, it allows the sieve weights to be supported, instead of on Polymath's truncated simplex, on a region in which single coordinates may be as large as about 0.380.38. On this region we use product trial functions multiplied by an optimised function of the sum of the coordinates, and we certify the resulting variational quotients in ball arithmetic, computing the relevant distribution functions by Fourier inversion.

Provenance statement

The manuscript, LaTeX source and ancillary computational materials were supplied by Avi Eisenberg. The accompanying materials include Python programs using python-flint/Arb, NumPy and SciPy for numerical certification, C programs for constructing and checking admissible tuples, input data, reproduction instructions and recorded computation logs. ChatGPT was used to review the manuscript. The submitter prepared the files for dissemination and checked that the LaTeX source compiles; this compilation check does not constitute independent verification of the mathematical proof.

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Version history

  1. v1Submitted by Annanay KapilaInitial depositCurrentOct 20, 2026