Note for the Large and Small Gaps

EasyChair Preprint 13449

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. On the one hand, 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, J{\'a}nos 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. On the other hand, the Andrica's conjecture deals with the difference between the square roots of consecutive prime numbers. While mathematicians have showed it true for a vast number of primes, a general solution remains elusive. We consider the inequality $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ for two successive prime numbers $p_{n}$ and $p_{n+1}$, where $\theta(x)$ is the Chebyshev function. In this note, under the assumption that the inequality $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ holds for all $n \geq 1.3002 \cdot 10^{16}$, we prove that the Andrica's conjecture is true. Since $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ holds indeed for large enough prime number $p_{n}$, then we show that the statement of the Andrica's conjecture can always be true for all primes greater than some threshold.

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

@booklet{EasyChair:13449,
  author    = {Frank Vega},
  title     = {Note for the Large and Small Gaps},
  howpublished = {EasyChair Preprint 13449},
  year      = {EasyChair, 2024}}