Giuseppe Giuga conjectured in 1950 that a natural number n is prime if and only if it satisfies the congruence 1n-1+2n-1+ ... + (n-1)n-1 = -1 mod n. Progress in validating or disproving the conjecture has been minimal, with the most significant advance being the knowledge that a counter-example would need at least 19,907 digits. To gain new insights into Giuga's conjecture, we explore it in the broader context of number fields. We present a generalized version of the conjecture and prove generalizations of many of the major results related to the conjecture. We introduce the concept of a Giuga ideal and perform computational searches for partial counter-examples to the generalized conjecture. We investigate the relationship between the existence of a counter-example in one number field with the existence of counter-examples in others, with a particular focus on quadratic extensions. This paper lays the preliminary foundation for answering the question: When does the existence of a counter-example in a number field imply the existence of a counter-example in the integers?
Burns, Jamaris; Casey, Katherine; Gichimu, Duncan; and Stinson, Kerrek
"Giuga's Primality Conjecture for Number Fields,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 18:
1, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/5