A search for Fibonacci-Wieferich and Wolstenholme primes

A prime p is called a Fibonacci-Wieferich prime if F_{p-({pover5})}equiv 0pmod{p^2} , where F_n is the n th Fibonacci number. We report that there exist no such primes p<2times10^{14} . A prime p is called a Wolstenholme prime if {2p-1\choose p-1}equiv 1pmod {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 1pmod {p^3} for all primes pge 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-({pover5})}/p modulo p .