Note On The Maximal Primes Gaps

Abstract

This note presents a result on the maximal prime gap of the form p_(n+1) - p_n <= C(log p_n)^(1+e), where C > 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently, the result shows that any short interval [x, x + y], y => C(log x)^(1+e), contains prime numbers for all sufficiently large real numbers x => x_0 unconditionally. An application demonstrates that a prime p => x > 2 can be determined in deterministic polynomial time O(log(x)^8).