manuscripta mathematica

, Volume 87, Issue 1, pp 501–510 | Cite as

On Giuga’s conjecture

  • Takashi Agoh


In this paper we shall investigate Giuga’s conjecture which asserts an interesting characterization of prime numbers, just as Wilson’s Theorem. Some variations and consequences of the Giuga congruence are discussed by means of Bernoulli numbers. In addition, we shall study various quotients relating to the integers satisfying the Giuga congruence.


Prime Factor Prime Number Euler Number Bernoulli Number Composite Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Agoh, On Bernoulli and Euler numbers, manuscripta math.61 (1988), 1–10zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    T. Agoh, On Fermat and Wilson quotients, Preprint (1995)Google Scholar
  3. 3.
    T. Agoh, K. Dilcher and L. Skula, Fermat and Wilson quotients for composite moduli, Preprint (1995)Google Scholar
  4. 4.
    W.R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math.140 (1994), 1–20CrossRefMathSciNetGoogle Scholar
  5. 5.
    E. Bedocchi, Nota ad uma congettura sui numeri primi, Riv. Mat. Univ. Parma11 (1985), 229–236zbMATHMathSciNetGoogle Scholar
  6. 6.
    D. Borwein, J.M. Borwein, P.B. Borwein and R. Girgensohn, Giuga’s conjecture on primality, Preprint (1994)Google Scholar
  7. 7.
    G. Giuga, Su una presumibile proprietà caratteristica dei numeri primi, Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat.83 (1950), 511–528MathSciNetGoogle Scholar
  8. 8.
    E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math.39 (1938), 350–360CrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Lerch, Zur Theorie des Fermatschen Quotientena p−1−1/p=q(a), Math. Annalen60 (1905), 471–490CrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Lerch Sur les théorèmes de Sylvester concernant le quotient de Fermat, C. R. Acad. Sci. Paris142 (1906), 35–38Google Scholar
  11. 11.
    P. Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988zbMATHGoogle Scholar
  12. 12.
    P. Ribenboim, The little book of big primes, Springer-Verlag, New York, 1991zbMATHGoogle Scholar
  13. 13.
    W. Sierpiński, Elementary number theory, Hafner, New York, 1964Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Takashi Agoh
    • 1
  1. 1.Department of MathematicsScience University of TokyoChibaJapan

Personalised recommendations