A Note on Oppermann's Conjecture

Frank Vega

doi:10.20944/preprints202408.1161.v4

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preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202408.1161.v4

Preprint Article Version 4 This version is not peer-reviewed

A Note on Oppermann's Conjecture

Frank Vega

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 3 : Received: 22 August 2024 / Approved: 23 August 2024 / Online: 23 August 2024 (09:45:01 CEST)
Version 4 : Received: 6 September 2024 / Approved: 12 September 2024 / Online: 12 September 2024 (10:56:11 CEST)

How to cite: Vega, F. A Note on Oppermann's Conjecture. Preprints 2024, 2024081161. https://doi.org/10.20944/preprints202408.1161.v4 Vega, F. A Note on Oppermann's Conjecture. Preprints 2024, 2024081161. https://doi.org/10.20944/preprints202408.1161.v4

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MDPI and ACS Style

Vega, F. A Note on Oppermann's Conjecture. Preprints 2024, 2024081161. https://doi.org/10.20944/preprints202408.1161.v4

APA Style

Vega, F. (2024). A Note on Oppermann's Conjecture. Preprints. https://doi.org/10.20944/preprints202408.1161.v4

Chicago/Turabian Style

Vega, F. 2024 "A Note on Oppermann's Conjecture" Preprints. https://doi.org/10.20944/preprints202408.1161.v4

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 square root function on inequalities involving prime numbers. This proof simultaneously establishes Andrica's, Legendre's, and Brocard's conjectures.

Keywords

prime gaps; prime numbers; square root; inequality

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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