Wilson quotients for composite moduli
HTML articles powered by AMS MathViewer
- by Takashi Agoh, Karl Dilcher and Ladislav Skula PDF
- Math. Comp. 67 (1998), 843-861 Request permission
Abstract:
An analogue for composite moduli $m \geq 2$ of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by $m$ is investigated; such an $m$ will be called a “Wilson number". It is shown that numbers in certain infinite classes cannot be Wilson numbers. Eight new Wilson numbers up to 500 million were found.References
- Takashi Agoh, On Bernoulli and Euler numbers, Manuscripta Math. 61 (1988), no. 1, 1–10. MR 939135, DOI 10.1007/BF01153577
- T. Agoh, K. Dilcher, and L. Skula, Fermat quotients for composite moduli, J. Number Theory 66 (1997), 29–50.
- Richard E. Crandall, Topics in advanced scientific computation, Springer-Verlag, New York; TELOS. The Electronic Library of Science, Santa Clara, CA, 1996. MR 1392472, DOI 10.1007/978-1-4612-2334-4
- Richard Crandall, Karl Dilcher, and Carl Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), no. 217, 433–449. MR 1372002, DOI 10.1090/S0025-5718-97-00791-6
- Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966. MR 0245499
- H. Dubner, Searching for Wilson primes, J. Recreational Math. 21 (1989), 19–20.
- R. H. Gonter and E. G. Kundert, All prime numbers up to 18,876,041 have been tested without finding a new Wilson prime, Preprint (1994).
- K. E. Kloss, Some number-theoretic calculations, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 335–336. MR 190057, DOI 10.6028/jres.069B.035
- M. Lerch, Zur Theorie des Fermatschen Quotienten $\frac {a^{p-1}-1}{p} = q(a)$, Math. Annalen 60 (1905), 471–490.
- E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350–360.
- Paulo Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080, DOI 10.1007/978-1-4684-9938-4
- Paulo Ribenboim, The little book of big primes, Springer-Verlag, New York, 1991. MR 1118843, DOI 10.1007/978-1-4757-4330-2
Additional Information
- Takashi Agoh
- Affiliation: Department of Mathematics, Science University of Tokyo, Noda, Chiba 278, Japan
- Email: agoh@ma.noda.sut.ac.jp
- Karl Dilcher
- Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
- Email: dilcher@mscs.dal.ca
- Ladislav Skula
- Affiliation: Department of Mathematics, Faculty of Science, Masaryk University, 66295 Brno, Czech Republic
- Email: skula@math.muni.cz
- Received by editor(s): January 23, 1995
- Received by editor(s) in revised form: May 22, 1996
- Additional Notes: The first author was supported in part by a grant of the Ministry of Education, Science and Culture of Japan. The second author’s research was supported by NSERC of Canada. Research of the third author was supported by the Grant Agency of the Czech Republic, “Number Theory, its Algebraic Aspects and its Relationship to Computer Science", No. 201/93/2/22.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp. 67 (1998), 843-861
- MSC (1991): Primary 11A07; Secondary 11B68
- DOI: https://doi.org/10.1090/S0025-5718-98-00951-X
- MathSciNet review: 1464140