Results for Wieferich Primes

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 $v^{p-1}\equiv 1p^2$ 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 $2^{p-1}-1\equiv 0p^2$. 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.