A search for Fibonacci-Wieferich and Wolstenholme primes

Abstract:

A prime $p$ is called a Fibonacci-Wieferich prime if $F_{p-({p\over 5})}\equiv 0\pmod {p^2}$, where $F_n$ is the $n$th Fibonacci number. We report that there exist no such primes $p<2\times 10^{14}$. A prime $p$ is called a Wolstenholme prime if ${2p-1\choose p-1}\equiv 1\pmod {p^4}$. To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes $p<10^9$. Wolstenholme, in 1862, proved that ${2p-1\choose p-1}\equiv 1\pmod {p^3}$ for all primes $p\ge 5$. It is estimated by a heuristic argument that the “probability” that $p$ is Fibonacci-Wieferich (independently: that $p$ is Wolstenholme) is about $1/p$. We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient $F_{p-({p\over 5})}/p$ modulo $p$.