# Agoh-Giuga conjecture

In 1950, Giuseppe Giuga conjectured that if and only if an integer $p$ is prime then it will satisfy the congruence

 $\sum_{i=1}^{p-1}i^{p-1}\equiv-1\mod p.$

This is sometimes called the Giuga conjecture. Takashi Agoh rephrased the conjecture as $pB_{p-1}\equiv-1\mod p$, where $B$ is a Bernoulli number; this is called the Agoh-Giuga conjecture. In 2003 Simon Plouffe performed an exhaustive search for a counterexample below 50000 but came up empty.

Title Agoh-Giuga conjecture AgohGiugaConjecture 2013-03-22 16:17:55 2013-03-22 16:17:55 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Conjecture msc 11D85 Giuga conjecture Giuga’s conjecture Agoh conjecture Agoh’s conjecture