A search for FibonacciWieferich and Wolstenholme primes
Abstract
A prime p is called a FibonacciWieferich 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 {2p1\choose p1}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 {2p1\choose p1}equiv 1pmod {p^3} for all primes pge 5 . It is estimated by a heuristic argument that the ``probability'' that p is FibonacciWieferich (independently: that p is Wolstenholme) is about 1/p . We provide some statistical data relevant to occurrences of small values of the FibonacciWieferich quotient F_{p({pover5})}/p modulo p .
 Publication:

Mathematics of Computation
 Pub Date:
 December 2007
 DOI:
 10.1090/S0025571807019552
 Bibcode:
 2007MaCom..76.2087M
 Keywords:

 Fibonacci number;
 Wieferich prime;
 WallSunSun prime;
 Wolstenholme prime.