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Journal of Interdisciplinary Mathematics

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502
CiteScore 2025: 1.9 (Q2)

CiteScore 2025: 1.9 (Q2)

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Journal of Interdisciplinary Mathematics
Open Access Research Article

A proof of cases of de Polignac’s conjecture

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pp. 1825–1835Vol. 28Issue 5August 2025DOI: 10.47974/JIM-2095XML
Received:
05 Jun 2024
Published Online:
13 Aug 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-2095
Pages:
1825–1835

Abstract

For n ≥ 1 let pn denote the nth prime number. Let  S = {1, 7, 11, 13, 17, 19, 23, 29}, the set of positive integers which are both less than and relatively prime to 30. For x ≥ 0, let Tx := {30x + i | i∈S}. For each x, Tx contains at most seven primes. Let [  ] denote the floor or greatest integer function. For each integer s ≥ 30 let π7 (s) denote the number of integers x, 0 ≤ x < [s/30] for which Tx contains seven primes. Let m ≥ 1010 be an integer and let PKm denote the largest prime number less than √Pmi=1 pi. In this paper we show that  Pmi=1 pi/8(Km+1) < p7 (Pmi=1 pi) and thereby prove that there are infinitely many values of x for which Tx contains seven primes. This, in particular, proves the well known twin prime conjecture as well as several cases of Alphonse de Polignac’s conjecture that for every even number k, there are infinitely many pairs of prime numbers p and p’ for which p’ – p = k.

Keywords

PrimesSeven prime setsSieve methods

Subject Classifications

11N0511N36

References

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