Results for Wieferich Primes

Abstract

Let $v\geq 2$ be a fixed integer, and let $x \geq 1$ and $z \geq x$ be large numbers. The exact asymptotic formula for the number of Wieferich primes $p$ such that $ v^{p-1} \equiv 1 \bmod p^2$ in the short interval $[x,x+z]$ is proposed in this note. The search conducted on the last 100 years have produced two primes $p<x=10^{15}$ such that $2^{p-1} -1\equiv 0 \bmod p^2$. The probabilistic and theoretical information within predicts the existence of another base $v=2$ prime on the interval $[10^{15},10^{40}]$. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demontrate that the subset of nonWieferich primes has density 1.