Note for the Prime Gaps

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. $g_{n}=p_{n+1}-p_{n}$. There isn't a verified solution to Andrica's conjecture yet. The conjecture itself 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.