A survey of results relating to Giuga’s conjecture on primality
Jonathan Michael Borwein, Eric W. M. Wong · CRM proceedings & lecture notes · 1997
This article is an expanded version of the talk given by the first author at the $25$th Anniversary Conference of the Centre de R\'echerches Math\'ematiques. In ~1950, G.~Giuga conjectured that if an integer~$n$ satisfies $\sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1$ mod~$n$, then~$n$ must be a prime. In this paper, we survey and complement a recent article (B$^{3}$G) on this interesting and now well established conjecture. \par Giuga proved that~$n$ is a counterexample to his conjecture if and only if each prime divisor~$p$ of~$n$ satisfies $(p-1) \mid (n/p-1)$ and $p \mid (n/p-1)$. Using this characterization, he proved computationally that any counterexample has at least 1000~digits; equipped with more computing power, E.~Bedocchi later raised this bound to 1700~digits. By improving on their method, we determine that any counterexample has at least 13,800~digits. \par We also give some new results on the second of the above conditions. This leads to open questions about what we call Giuga numbers and Giuga sequences. Finally we study eight natural variations on our theme. Complete details are provided in this section since the results are entirely new.