关于Wieferich Primes
Results for Wieferich Primes
摘要：
令$v\geq 2$为固定整数，令$x \geq 1$和$z \geq x$为大数。本文给出了在短区间$[x,x+z]$内满足$ v^{p-1} \equiv 1 \bmod p^2$的Wieferich素数$p$个数的精确渐近公式.在过去的100年里进行的搜索已经产生了两个素数$p<x=10^{15}$，使得$2^{p-1} -1\equiv 0 \bmod p^2$。其中的概率和理论信息预测了间隔$[10^{15},10^{40}]$上另一个基$v=2$素数的存在。此外，利用关于Wieferich素数个数上界的一个结果证明了非Wieferich素数子集的密度为1.
英文摘要：
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.
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