Results for Wieferich Primes

TL;DR

This paper derives an asymptotic formula for counting Wieferich primes within short intervals and discusses their distribution, including the rarity of such primes and predictions for their occurrence in large ranges.

Contribution

It provides the first asymptotic estimate for Wieferich primes in short intervals and analyzes their density and distribution properties.

Findings

Only two Wieferich primes found below 10^15.

Predicted existence of another Wieferich prime between 10^15 and 10^40.

Non-Wieferich primes have density 1.

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.