A search for Fibonacci-Wieferich and Wolstenholme primes

Authors:
Publisher website
Google Scholar
Abstract
A prime p p is called a Fibonacci-Wieferich prime if F p − ( p 5 ) ≡ 0 ( mod p 2 ) F_{p-({p\over 5})}\equiv 0\pmod {p^2} , where F n F_n is the n n th Fibonacci number. We report that there exist no such primes p > 2 × 10 14 p>2\times 10^{14} . A prime p p is called a Wolstenholme prime if ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ) {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 p>10^9 . Wolstenholme, in 1862, proved that ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {2p-1\choose p-1}\equiv 1\pmod {p^3} for all primes p ≥ 5 p\ge 5 . It is estimated by a heuristic argument that the “probability” that p p is Fibonacci-Wieferich (independently: that p p is Wolstenholme) is about 1 / p 1/p . We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient F p − ( p 5 ) / p F_{p-({p\over 5})}/p modulo p p .

This publication has 16 references indexed in Scilit: