Small gaps between almost-twin primes

Bin Chen

doi:10.1515/forum-2024-0036

Article

Small gaps between almost-twin primes

Bin Chen

Published/Copyright: July 16, 2025

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From the journal Forum Mathematicum

Abstract

Let m ∈ N be large. We show that there exist infinitely many primes q 1 < ⋯ < q m + 1 such that

q m + 1 − q 1 = O ⁢ ( e 7.63 ⁢ m )

and q j + 2 has at most 7.36 ⁢ m log ⁡ 2 + 4 ⁢ log ⁡ m log ⁡ 2 + 21 prime factors for each 1 ≤ j ≤ m + 1 . This improves the previous result of Li and Pan, replacing m 4 ⁢ e 8 ⁢ m by e 7.63 ⁢ m and 16 ⁢ m log ⁡ 2 + 5 ⁢ log ⁡ m log ⁡ 2 + 37 by 7.36 ⁢ m log ⁡ 2 + 4 ⁢ log ⁡ m log ⁡ 2 + 21 . The main inputs are the Maynard–Tao sieve, a minorant for the indicator function of the primes constructed by Baker and Irving, for which a stronger equidistribution theorem in arithmetic progressions to smooth moduli is applicable, and Tao’s approach previously used to estimate ∑ x ≤ n < 2 ⁢ x 1 P ⁢ ( n ) ⁢ 1 P ⁢ ( n + 12 ) ⁢ ω n , where 1 P stands for the characteristic function of the primes and ω n are multidimensional sieve weights.

Keywords: Maynard–Tao sieve; Tao’s approach; minorant for the indicator function of the primes; prime gaps

MSC 2020: 11N05; 11N35; 11N36

Funding statement: The work was supported by the China Scholarship Council (CSC).

Acknowledgements

The author would like to thank to thank Terence Tao and Gregory Debruyne for many useful conversations and suggestions.

Communicated by: Guozhen Lu

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Received: 2024-01-16

Revised: 2024-07-31

Published Online: 2025-07-16

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https://doi.org/10.1515/forum-2024-0036

Keywords for this article

Maynard–Tao sieve; Tao’s approach; minorant for the indicator function of the primes; prime gaps