Giuga's Conjecture on Primality

G. Giuga conjectured that if an integer n satisses n?1 P k=1 k n?1 ?1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisses (p ? 1) j (n=p ? 1) and p j (n=p ? 1). Using this characterization, he proved computationally that any counterexample has at least 1,000 digits; equipped with more computing power, E. Bedocchi later raised this bound to 1,700 digits. By improving on their method, we determine that any counterexample has at least 13,800 digits. We also give some new results on the second of the above conditions. This leads, in our opinion, to some interesting questions about what we call Giuga numbers and Giuga sequences.