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On Giuga’s conjecture. (English) Zbl 0845.11004

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

${1}^{-1}+{2}^{n-1}+\cdots +{\left(n-1\right)}^{n-1}\equiv -1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}n\right)·\phantom{\rule{2.em}{0ex}}\left(*\right)$

Since for prime numbers $n$ $\left(*\right)$ 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.

##### MSC:
 11A07 Congruences; primitive roots; residue systems 11A41 Elementary prime number theory
##### References:
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