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October 16, 2025Open Access

Rough numbers between consecutive primes

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Authors

AGAyla GafniTTTerence Tao

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Overview

This paper uncovers the presence of specific natural numbers in gaps between consecutive primes, highlighting patterns indicating predictions of Erdős.

Key Points

  • Almost all gaps between consecutive primes contain a number whose least prime factor is at least the length of the gap.
  • The number of exceptional gaps within [X, 2X] is shown to be at most O(X/log² X), confirming Erdős' predictions.
  • Under certain assumptions, an explicit asymptotic formula for exceptional gaps is established, indicating cX/log² X for some constant c.
  • The findings significantly rely on refined techniques for singular series developed by Montgomery and Soundararajan.

Cite This Study

Gafni et al. (2025) studied this question.

synapsesocial.com/papers/68f12bfb2107091eab27a372https://doi.org/10.48550/arxiv.2508.06463
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