## A search for Fibonacci-Wieferich and Wolstenholme primes

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- by Richard J. McIntosh and Eric L. Roettger;
- Math. Comp.
**76**(2007), 2087-2094 - DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
- Published electronically: April 17, 2007
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## 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 $p$.

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## Bibliographic Information

**Richard J. McIntosh**- Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
- Email: mcintosh@math.uregina.ca
**Eric L. Roettger**- Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Email: roettgee@math.ucalgary.ca
- Received by editor(s): June 14, 2005
- Received by editor(s) in revised form: May 19, 2006
- Published electronically: April 17, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 2087-2094 - MSC (2000): Primary 11A07, 11A41, 11B39, 11Y99
- DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
- MathSciNet review: 2336284