Ukrainian Mathematical Journal

, Volume 67, Issue 11, pp 1778–1785 | Cite as

Variations on Giuga Numbers and Giuga’s Congruence

  • J.-M. Grau
  • A. M. Oller-Marcén
Article
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A k -strong Giuga number is a composite integer such that ∑j = 1n − 1jn − 1≡ − 1 (mod n). We consider the congruence ∑j = 1n − 1jk(n − 1)≡ − 1 (mod n) for each k\( \epsilon \) ℕ (thus extending Giuga’s ideas for k = 1). In particular, it is proved that a pair (n, k) with composite n satisfies this congruence if and only if n is a Giuga number and ⋋(n) | k(n − 1). In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying ⋋(n) | k(n − 1).

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • J.-M. Grau
    • 1
  • A. M. Oller-Marcén
    • 2
  1. 1.Universidad de OviedoOviedoSpain
  2. 2.Centro Universitario de la DefensaZaragozaSpain

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