## You are here

HomeAgoh-Giuga conjecture

## Primary tabs

# 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.

Synonym:

Giuga conjecture, Giuga's conjecture, Agoh conjecture, Agoh's conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

11D85*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections