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

Conference paper
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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)

## References

1. 1.
T. Agoh, K. Dilcher, L. Skula, Wilson quotients for composite moduli. Math. Comp. 67, 843–861 (1998)
2. 2.
J.H. Conway, R.K. Guy, The Book of Numbers (Springer, New York, 1996)
3. 3.
J.B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson theorem. Integers 8, article A39 (2008)Google Scholar
4. 4.
E. Costa, R. Gerbicz, D. Harvey, A search for Wilson primes (2012, preprint); available at http://arxiv.org/abs/1209.3436
5. 5.
R. Crandall, K. Dilcher, C. Pomerance, A search for Wieferich and Wilson primes. Math. Comp. 66, 433–449 (1997)
6. 6.
P.J. Davis, Are there coincidences in mathematics? Am. Math. Mon. 88, 311–320 (1981)
7. 7.
L.E. Dickson, History of the Theory of Numbers, vol. 1. (Carnegie Institution of Washington, Washington, D.C. 1919); reprinted by (Dover, Mineola, NY 2005)Google Scholar
8. 8.
K. Dilcher, A bibliography of Bernoulli numbers (2011); available at http://www.mscs.dal.ca/~dilcher/bernoulli.html
9. 9.
J.B. Dobson, On Lerch’s formula for the Fermat quotient (2012, preprint); available at http://arxiv.org/abs/1103.3907
10. 10.
F.G. Dorais, D.W. Klyve, A Wieferich prime search up to 6. 7 × 1015. J. Integer Seq. 14, Article 11.9.2 (2011); available at http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html
11. 11.
F. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definiert werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. zu Berlin 36–42 (1850); reprinted in vol. 2 (Mathematische Werke, Chelsea, New York 1975) pp. 705–711Google Scholar
12. 12.
J.W.L. Glaisher, A congruence theorem relating to Eulerian numbers and other coefficients. Proc. Lond. Math. Soc. 32, 171–198 (1900)
13. 13.
GMP: The GNU Multiple Precision Arithmetic Library (2011); available at http://gmplib.org/
14. 14.
K. Goldberg, A table of Wilson quotients and the third Wilson prime. J. Lond. Math. Soc. 28, 252–256 (1953)
15. 15.
R.K. Guy, The strong law of small numbers. Am. Math. Mon. 95, 697–712 (1988)
16. 16.
R.K. Guy, Unsolved Problems in Number Theory, 3rd edn. (Springer, New York 2004)
17. 17.
G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Oxford University Press, Oxford 1989)Google Scholar
18. 18.
Klaster Instytutu Fizyki Teoretycznej UWr (2012); http://zero.ift.uni.wroc.pl/
19. 19.
D. Knuth, Johann Faulhaber and sums of powers. Math. Comp. 61, 277–294 (1993)
20. 20.
E. Lehmer, A note on Wilson’s quotient. Am. Math. Mon. 44, 237–238 (1937)
21. 21.
M. Lerch, Zur Theorie des Fermatschen Quotienten $$\frac{a^{p-1}-1} {p} = q(a)$$. Math. Ann. 60, 471–490 (1905)
22. 22.
D. MacHale, Comic Sections: The Book of Mathematical Jokes, Humour, Wit and Wisdom (Boole Press, Dublin 1993)Google Scholar
23. 23.
P.L. Montgomery, New solutions of $$a^{\,p-1} \equiv 1\pmod p^{\,2}$$. Math. Comp. 61, 361–363 (1993)
24. 24.
PARI/GP (2011); available at http://pari.math.u-bordeaux.fr/
25. 25.
P. Ribenboim, 1093. Math. Intelligencer 5, 28–34 (1983)Google Scholar
26. 26.
P. Ribenboim, The Book of Prime Number Records, 2nd. edn. (Springer, New York, 1989)Google Scholar
27. 27.
P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (Springer, New York, 2000)
28. 28.
J.H. Silverman, Wieferich’s criterion and the abc-conjecture. J. Number Theor. 30(2), 226–237 (1988)
29. 29.
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (2011); published at http://oeis.org/
30. 30.
J. Sondow, K. MacMillan, Reducing the Erdős-Moser equation $$1^{n} + 2^{n} + \cdots + k^{n} = (k + 1)^{n}$$ modulo k and k 2. Integers 11, article A34 (2011); expanded version available at http://arxiv.org/abs/1011.2154v1
31. 31.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers (Penguin Books, London, 1986)Google Scholar
32. 32.
Wikipedia: Wieferich prime (2012); http://en.wikipedia.org/wiki/Wieferich_prime

## Copyright information

© Springer Science+Business Media New York 2014

## Authors and Affiliations

1. 1.New YorkUSA