Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often

Joel E. Cohen and Dexter Senft
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 494–503
DOI: 10.7546/nntdm.2025.31.3.494-503
Full paper (PDF, 209 Kb)

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Authors and affiliations

Joel E. Cohen
1 Laboratory of Populations, The Rockefeller University
1230 York Avenue, Box 20, New York, NY 10065, USA
2  Department of Statistics, Columbia University, New York, NY
3  Department of Statistics, University of Chicago, Chicago, IL

Dexter Senft
4 Asymptotic Systems
Saratoga Springs, NY, USA

Abstract

The infinite sequence of gaps (first differences) between successive odd composite numbers contains only the numbers 2, 4, and 6. We prove that, for any natural number k, the sequence of gaps contains infinitely many k-tuplets of consecutive gaps all equal to 2. Infinitely many gaps equal 4. The sequence of gaps includes infinitely many gap pairs (4, 4) if the sequence of positive primes has infinitely many pairs of successive primes that differ by 4 (cousin primes), which is unproved but holds under a conjecture of Hardy and Littlewood. Gap triplets (4, 4, 4) never occur. Infinitely many gaps equal 6 if and only if there are infinitely many twin primes. Moreover, gap pairs (6, 6) occur infinitely often if other conjectures of Hardy and Littlewood are true. Six of the 27 potential triplets of values of gaps between successive odd composite numbers never occur: (4, 4, 4), (6, 6, 6), (6, 4, 4), (4, 4, 6), (6, 2, 6), and (6, 4, 6).

Keywords

  • Composite number
  • Cousin primes
  • Gaps
  • Odd number
  • Odd composite number
  • Prime number
  • Twin primes

2020 Mathematics Subject Classification

  • 11B83
  • 11A41

References

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Manuscript history

  • Received: 16 February 2025
  • Revised: 11 August 2025
  • Accepted: 13 August 2025
  • Online First: 15 August 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Cohen, J. E., & Senft, D. (2025). Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often. Notes on Number Theory and Discrete Mathematics, 31(3), 494-503, DOI: 10.7546/nntdm.2025.31.3.494-503.

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