Note for the Small Gaps

EasyChair Preprint 13353, version 2

Abstract

A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted $g_{n}$ is the difference between the (n + 1)st and the nth prime numbers, i.e. $g_{n}=p_{n+1}-p_{n}$. A twin prime is a prime that has a prime gap of two. The twin prime conjecture states that there are infinitely many twin primes. There isn't a verified solution to twin prime conjecture yet. In this note, using the Chebyshev function, we prove that $$\liminf_{n\to \infty }{\frac {g_{n}+g_{n-1}}{\log (p_{n}) + \log (p_{n} + 2)}} \geq 1,$$ under the assumption that the twin prime conjecture is false. It is well-known the proof of Daniel Goldston, János Pintz and Cem Yildirim which implies that $\liminf_{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0$. In this way, we reach an intuitive contradiction. Consequently, by reductio ad absurdum, we can conclude that the twin prime conjecture is true.

Keyphrases: Chebyshev function, Primorial numbers, prime gaps, prime numbers

@booklet{EasyChair:13353,
  author    = {Frank Vega},
  title     = {Note for the Small Gaps},
  howpublished = {EasyChair Preprint 13353},
  year      = {EasyChair, 2024}}