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A Note on Oppermann's Conjecture
Version 1
: Received: 14 August 2024 / Approved: 15 August 2024 / Online: 16 August 2024 (03:00:22 CEST)
Version 2 : Received: 18 August 2024 / Approved: 20 August 2024 / Online: 21 August 2024 (04:30:04 CEST)
Version 2 : Received: 18 August 2024 / Approved: 20 August 2024 / Online: 21 August 2024 (04:30:04 CEST)
How to cite: Vega, F. A Note on Oppermann's Conjecture. Preprints 2024, 2024081161. https://doi.org/10.20944/preprints202408.1161.v2 Vega, F. A Note on Oppermann's Conjecture. Preprints 2024, 2024081161. https://doi.org/10.20944/preprints202408.1161.v2
Abstract
A prime gap is the difference between consecutive prime numbers. The $n^{\text{th}}$ prime gap, denoted $g_{n}$, is calculated by subtracting the $n^{\text{th}}$ prime from the $(n+1)^{\text{th}}$ prime: $g_{n}=p_{n+1}-p_{n}$. Oppermann's conjecture is a prominent unsolved problem in pure mathematics concerning prime gaps. Despite verification for numerous primes, a general proof remains elusive. If true, the conjecture implies that prime gaps grow at a rate bounded by $g_{n}<{\sqrt {p_{n}}}$. This note presents a proof of Oppermann's conjecture using the Euler-Maclaurin formula on harmonic numbers. This proof simultaneously establishes Andrica's, Legendre's, and Brocard's conjectures.
Keywords
prime gaps; prime numbers; Euler-Maclaurin formula; harmonic numbers
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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