Agoh-Giuga conjecture

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

$$\sum _{i=1}^{p-1}{i}^{p-1}\equiv -1modp.$$

This is sometimes called the Giuga conjecture. Takashi Agoh rephrased the conjecture as $p{B}_{p-1}\equiv -1modp$, 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
Canonical name	AgohGiugaConjecture
Date of creation	2013-03-22 16:17:55
Last modified on	2013-03-22 16:17:55
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 11D85
Synonym	Giuga conjecture
Synonym	Giuga’s conjecture
Synonym	Agoh conjecture
Synonym	Agoh’s conjecture