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^{}
$$\sum _{a=1}^{p-1}{q}_{p}(a)\equiv {w}_{p}\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$$ |
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 http://arxiv.org/pdf/1110.3113v3.pdfarXiv.
Title | Fermat quotient |
---|---|
Canonical name | FermatQuotient |
Date of creation | 2013-03-22 19:34:22 |
Last modified on | 2013-03-22 19:34:22 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11A51 |
Classification | msc 11A41 |
Defines | Lerch’s formula |
Defines | Fermat–Wilson quotient |