preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202408.1161.v3

Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

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)

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 $g_{n}<{\sqrt {p_{n}}}$. We examine the ratio of Chebyshev functions for consecutive primes, $\frac{\theta(p_{n+1})}{\theta(p_{n})}$, and compare it to the square root of their ratio, $\sqrt {\frac{p_{n+1}}{p_{n}}}$. Assuming this inequality holds for primes larger than $10^{8}$, we demonstrate the truth of Oppermann's conjecture. For sufficiently large $n$, this inequality guarantees that $g_{n}<{\sqrt {p_{n}}}$ for all primes beyond a certain point. Our proof also validates Andrica's, Legendre's, and Brocard's conjectures for primes exceeding a specific threshold.

prime gaps; prime numbers; Chebyshev function; primorial numbers

Computer Science and Mathematics, Algebra and Number Theory

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