## You are here

HomeFermat quotient

## Primary tabs

# Fermat quotient

If $a$ is an integer not divisible by a positive prime $p$, then Fermat’s little theorem (a.k.a. Fermat’s theorem) guarantees that the difference $a^{{p-1}}\!-\!1$ is divisible by $p$. The integer

$q_{p}(a)\;:=\;\frac{a^{{p-1}}\!-\!1}{p}$ |

is called the Fermat quotient of $a$ modulo $p$. Compare it with the Wilson quotient $w_{p}$, which is similarly related to Wilson’s theorem.

Lerch’s formula

$\sum_{{a=1}}^{{p-1}}q_{p}(a)\;\equiv\;w_{p}\;\,\;\;(\mathop{{\rm mod}}p)$ |

for an odd prime $p$ connects the Fermat quotients and the Wilson quotient.

If $p$ is a positive prime but not a Wilson prime, and $w_{p}$ is its Wilson quotient, then the expression

$q_{p}(w_{p})\;=\;\frac{w_{p}^{{p-1}}\!-\!1}{p}$ |

is called the Fermat–Wilson quotient of $p$. Sondow proves in [1] that the greatest common divisor of all Fermat–Wilson quotients is 24.

# References

- 1
Jonathan Sondow:
*Lerch Quotients, Lerch Primes, Fermat–Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771*. Available at arXiv.

Defines:

Lerch's formula, Fermat--Wilson quotient

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11A51*no label found*11A41

*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