On Primes In Short Intervals

TL;DR

This paper presents an elementary proof for the existence of primes within short intervals of a certain size and extends Bertrand's postulate to arithmetic progressions, contributing to understanding prime distribution.

Contribution

It provides an unconditional elementary argument for primes in short intervals and extends Bertrand's postulate to arithmetic progressions.

Findings

Primes exist in intervals [x, x + y] with y >= x^(1/2)(log x)^e

Elementary proof does not rely on deep analytic methods

Extension of Bertrand's postulate to arithmetic progressions

Abstract

This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently large number x > 0. Further, an extension of Bertrand's postulate to arithmetic progressions will be considered