The gap $g_n$ between two consecutive primes satisfies $g_n=O({p_n}^{2/3})$

Abstract

The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$.