Results for Wieferich Primes

Abstract

Let $v\ge 2$ be a fixed integer, and let $x\ge 1$ and $z\ge x$ be large numbers. The exact asymptotic formula for the number of Wieferich primes $p$ such that $v^{p-1}\equiv 1\mathrm{mod}{p}^2$ 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 $2^{p-1}-1\equiv 0\mathrm{mod}{p}^2$. 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.

Fifty Three Pages. Keywords: Distribution of Prime, Wieferiech prime, Finite Ring

References in corpus (1)

• Character sums for elliptic curve densities