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?

Author Bio

Jamaris Burns graduated from Johnson C. Smith University in 2015 with a B.S. in Mathematics. She worked on this research while in the SUAMI 2013 cohort at Carnegie Mellon University. She is currently pursuing her Masters degree in Data Analytics and working as a Project Management Specialist for a data and technology management company. Other interests of hers are painting, writing poetry and traveling.

Katherine Casey graduated in 2014 with a B.A. in mathematics from Cornell University with honors as a Rawlings Cornell presidential research scholar. She worked on this project in 2013 while at Carnegie Mellon University's SUAMI. She presented part of this work at the student poster session at JMM 2014. She has since graduated from the University of Oxford with an MSc in Mathematics and Foundations of Computer Science. In her spare time, she loves to sing classical choral and vocal works.

Duncan Gichimu studied Mathematics And Economics at Towson University. He worked on this research at Carnegie Mellon Unveristiy's SUAMI in 2015. His hobbies include calisthenics, reading and programming. His favorite programming language is Python and he is especially intrigued by algebraic data types in Haskell.

Kerrek Stinson is presently a graduate student at Carnegie Mellon University focusing on applied analysis. His work on this paper was completed during his time as an undergraduate at CMU's SUAMI in 2015. Some of his hobbies include, but are not limited to, math.