Wilson Quotients for Composite Moduli

Authors:
Takashi Agoh, Karl Dilcher and Ladislav Skula

Journal:
Math. Comp. **67** (1998), 843-861

MSC (1991):
Primary 11A07; Secondary 11B68

MathSciNet review:
1464140

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An analogue for composite moduli of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by is investigated; such an 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.

**1.**Takashi Agoh,*On Bernoulli and Euler numbers*, Manuscripta Math.**61**(1988), no. 1, 1–10. MR**939135**, 10.1007/BF01153577**2.**T. Agoh, K. Dilcher, and L. Skula,*Fermat quotients for composite moduli*, J. Number Theory**66**(1997), 29-50.**3.**Richard E. Crandall,*Topics in advanced scientific computation*, Springer-Verlag, New York; TELOS. The Electronic Library of Science, Santa Clara, CA, 1996. MR**1392472****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**, 10.1090/S0025-5718-97-00791-6**5.**Leonard Eugene Dickson,*History of the theory of numbers. Vol. I: Divisibility and primality.*, Chelsea Publishing Co., New York, 1966. MR**0245499****6.**H. Dubner,*Searching for Wilson primes*, J. Recreational Math.**21**(1989), 19-20.**7.**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).**8.**K. E. Kloss,*Some number-theoretic calculations*, J. Res. Nat. Bur. Standards Sect. B**69B**(1965), 335–336. MR**0190057****9.**M. Lerch,*Zur Theorie des Fermatschen Quotienten*, Math. Annalen**60**(1905), 471-490.**10.**E. Lehmer,*On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson*, Ann. of Math.**39**(1938), 350-360.**11.**Paulo Ribenboim,*The book of prime number records*, Springer-Verlag, New York, 1988. MR**931080****12.**Paulo Ribenboim,*The little book of big primes*, Springer-Verlag, New York, 1991. MR**1118843**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11A07,
11B68

Retrieve articles in all journals with MSC (1991): 11A07, 11B68

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

DOI:
http://dx.doi.org/10.1090/S0025-5718-98-00951-X

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.\endgraf The second author’s research was supported by NSERC of Canada.\endgraf 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.

Article copyright:
© Copyright 1998
American Mathematical Society