Legendre Conjecture over Arithmetic Progressions

Abstract

Let $1\le a<q$ be a pair of small integers such that $\gcd (a,q)=1$ and let $x>1$ be a large number. This note discusses the existence of a short sequence of primes $p\equiv a\mathrm{mod}q$ between two squares $x^2$ and $(x+1{)}^2$.

arXiv commentSeven Pages. Keywords: Prime in short interval; Prime in arithmetic progression; Distribution of prime; Legendre conjecture