Frangible prime gaps — Draconitic Research

Abstract

A prime gap (p_n, p_n+1) is \textit{frangible} if the smallest prime divisor of every integer m with p_n<m<p_n+1 is strictly less than the size of the gap, p_n+1−p_n. Gafni and Tao recently proved that the number of frangible gaps up to X is ≪ X/log² X, answering a question of Erdős. They also showed that under the Dickson--Hardy--Littlewood prime tuples conjecture this number is ∼ c X/log² X for a constant c, for which they obtained a rough estimate. We describe improved methods for estimating this constant under the same hypothesis, and establish that its value lies in $[2.7416, 2.7984]$, confirming a speculation of Gafni and Tao. In addition, we present the results of a complete survey of frangible gaps up to 10¹⁷. An ensuing empirical measurement of c matches our estimate very well.

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BibTeX RIS

BibTeX

@misc{mossinghoff2026frangible,
  title = {{Frangible prime gaps}},
  author = {Michael J. Mossinghoff and Timothy S. Trudgian},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34525},
  eprint = {2610.34525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34525},
}

RIS

TY  - UNPB
TI  - Frangible prime gaps
AU  - Michael J. Mossinghoff
AU  - Timothy S. Trudgian
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34525
UR  - https://arxiv.org/abs/2610.34525
AB  - A prime gap (p_n, p_n+1) is \textit{frangible} if the smallest prime divisor of every integer m with p_n<m<p_n+1 is strictly less than the size of the gap, p_n+1−p_n. Gafni and Tao recently proved that the number of frangible gaps up to X is ≪ X/log^2 X, answering a question of Erdős. They also showed that under the Dickson--Hardy--Littlewood prime tuples conjecture this number is ∼ c X/log^2 X for a constant c, for which they obtained a rough estimate. We describe improved methods for estimating this constant under the same hypothesis, and establish that its value lies in $[2.7416, 2.7984]$, confirming a speculation of Gafni and Tao. In addition, we present the results of a complete survey of frangible gaps up to 10^17. An ensuing empirical measurement of c matches our estimate very well.
ER  -

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