A k -strong Giuga number is a composite integer such that ∑ n − 1 j = 1 j n − 1≡ − 1 (mod n). We consider the congruence ∑ n − 1 j = 1 j k(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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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 11, pp. 1573–1578, November, 2015.
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Grau, JM., Oller-Marcén, A.M. Variations on Giuga Numbers and Giuga’s Congruence. Ukr Math J 67, 1778–1785 (2016). https://doi.org/10.1007/s11253-016-1189-2
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DOI: https://doi.org/10.1007/s11253-016-1189-2