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Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-Non-Wilson Primes 2, 3, 14771

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 101)

Abstract

The Fermat quotient \(q_{p}(a):= (a^{p-1} - 1)/p\), for prime p∤ a, and the Wilson quotient \(w_{p}:= ((p - 1)! + 1)/p\) are integers. If pw p , then p is a Wilson prime. For odd p, Lerch proved that \((\sum\nolimits_{a=1}^{p-1}q_{p}(a) - w_{p})/p\) is also an integer; we call it the Lerch quotient  p . If p p we say p is a Lerch prime. A simple Bernoulli number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to 3 × 106; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if p is a non-Wilson prime, then \(q_{p}(w_{p})\) is an integer that we call the Fermat-Wilson quotient of p. The GCD of all \(q_{p}(w_{p})\) is shown to be 24. If \(p\mid q_{p}(a)\), then p is a Wieferich prime base a; we give a survey of them. Taking a = w p , if \(p\mid q_{p}(w_{p})\) we say p is a Wieferich-non-Wilson prime. There are three up to 107, namely, 2, 3, 14771. Several open problems are discussed.

Keywords

Fermat quotient Wilson quotient Wilson prime Lerch’s formula Bernoulli number Faulhaber’s formula von Staudt-Clausen theorem Glaisher’s congruence E. Lehmer’s test Mathematica Wieferich prime abc-conjecture 

MSC 2010:

11A41 (primary) 11B68 (secondary) 

Notes

Acknowledgements

I am grateful to Wadim Zudilin for suggestions on the test, for a simplification in computing WW primes, and for verifying that there are no new ones up to 30000, using PARI/GP [24]. I thank Marek Wolf for computing Lerch primes and Michael Mossinghoff for computing WW primes.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.New YorkUSA

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