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Zbl 0845.11004
Agoh, Takashi
On Giuga's conjecture. (English)
[J] Manuscr. Math. 87, No.4, 501-510 (1995). ISSN 0025-2611; ISSN 1432-1785/e

{\it G. Giuga} [Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. 83, 511-518 (1951; Zbl 0045.01801)] conjectured that no composite number $n$ satisfies the congruence $$1^{- 1}+ 2^{n- 1}+\cdots+ (n- 1)^{n- 1}\equiv -1\pmod n.\tag{$*$}$$ Since for prime numbers $n$ $(*)$ obviously holds, the truth of Giuga's conjecture would provide a characterization of primes. In the present paper, the author discusses consequences and variations of the congruence above. He points out some relations to Bernoulli numbers and Euler, Fermat and Wilson quotients.
[T.Maxsein (Clausthal)]

MSC 2000:: *11A07 Congruences, etc.
11A41 Elementary prime number theory

Keywords: Carmichael numbers; Euler quotients; Fermat quotients; Giuga's conjecture; characterization of primes; congruence; Bernoulli numbers; Wilson quotients

Citations: Zbl 0045.01801

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