Variations on Giuga Numbers and Giuga’s Congruence

• Date: 2015
• Main Authors: Grau, José María, Грау, Хосе Марія
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2015
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/2091
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

A $k$ -strong Giuga number is a composite integer such that $∑_{j = 1}^{n − 1} j^{n − 1} ≡  − 1 (mod n)$. We consider the congruence $∑_{j = 1}^{n − 1} j^{k(n − 1)} ≡  − 1 (mod n)$ for each $k ϵ ℕ$ (thus extending Giuga’s ideas for $k = 1$). In particular, it is proved that a pair $(n, k)$ with comp...