On Giuga's conjecture
type: Item
instance of: scholarly article
title: On Giuga's conjecture
zbMATH Open document ID: 0845.11004
DOI: 10.1007/BF02570490
author: Takashi Agoh
published in: Manuscripta Mathematica
publication date: 1995-12-10T00:00:00Z
full work available at URL: https://eudml.org/doc/156103
review text: \textit{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.
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 824175
zbMATH Keywords: Carmichael numbers
zbMATH Keywords: Euler quotients
zbMATH Keywords: Fermat quotients
zbMATH Keywords: Giuga's conjecture
zbMATH Keywords: characterization of primes
zbMATH Keywords: congruence
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Wilson quotients
reviewed by: Thomas Maxsein
MaRDI profile type: Publication
Wikidata QID: Q56455766
cites work: On Bernoulli and Euler numbers
cites work: Q4873235
cites work: Wilson quotients for composite moduli
cites work: A regularized Siegel-Weil formula: The first term identity
cites work: Giuga's Conjecture on Primality
cites work: Q5808457
cites work: On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson
cites work: Q4039755
cites work: Q4001553
description: scientific article; zbMATH DE number 824175