Fermat Quotient

In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as

:$q_p(a) = \frac,$

or

:$\delta_p(a) = \frac$.

This article is about the former; for the latter see ''p''-derivation. The quotient is named after Pierre de Fermat.

If the base ''a'' is coprime to the exponent ''p'' then Fermat's little theorem says that ''q''_''p''(''a'') will be an integer. If the base ''a'' is also a generator of the multiplicative group of integers modulo ''p'', then ''q''_''p''(''a'') will be a cyclic number, and ''p'' will be a full reptend prime.

Properties

From the definition, it is obvious that

:$\begin q_p(1) &\equiv 0 && \pmod \\ q_p(-a)&\equiv q_p(a) && \pmod\quad (\text 2 \mid p-1) \end$

In 1850, Gotthold Eisenstein proved that if ''a'' and ''b'' are both coprime to ''p'', then:

:$\begin q_p(ab) &\equiv q_p(a)+q_p(b) &&\pmod \\ q_p(a^r) &\equiv rq_p(a) &&\pmod \\ q_p(p \mp a) &\equiv q_p(a) \pm \tfrac &&\pmod \\ q_p(p \mp 1) &\equiv \pm 1 && \pmod \end$

Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply

:$\begin q_p \!\left(\tfrac \right) &\equiv -q_p(a) && \pmod \\ q_p \!\left(\tfrac \right) &\equiv q_p(a) - q_p(b) &&\pmod \end$

In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:

:$q_p(a+np)\equiv q_p(a)-n\cdot\tfrac \pmod.$

From this, it follows that:

:$q_p(a+np^2)\equiv q_p(a) \pmod.$

Lerch's formula

M. Lerch proved in 1905 that

:$\sum_^q_p(j)\equiv W_p\pmod.$

Here $W_p$ is the Wilson quotient.

Special values

Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of the reciprocals modulo ''p'' of the numbers lying in the first half of the range :

:$-2q_p(2) \equiv \sum_^ \frac \pmod.$

Later writers showed that the number of terms required in such a representation could be reduced from 1/2 to 1/4, 1/5, or even 1/6:

:$-3q_p(2) \equiv \sum_^ \frac \pmod.$

:$4q_p(2) \equiv \sum_^ \frac + \sum_^ \frac \pmod.$

:$2q_p(2) \equiv \sum_^ \frac \pmod.$

Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:

:$-3q_p(3) \equiv 2\sum_^ \frac \pmod.$

:$-5q_p(5) \equiv 4\sum_^ \frac + 2\sum_^ \frac \pmod.$

Generalized Wieferich primes

If ''q''_''p''(''a'') ≡ 0 (mod ''p'') then ''a''^''p''−1 ≡ 1 (mod ''p''²). Primes for which this is true for ''a'' = 2 are called Wieferich primes. In general they are called ''Wieferich primes base a.'' Known solutions of ''q''_''p''(''a'') ≡ 0 (mod ''p'') for small values of ''a'' are:

:

For more information, see and.Wieferich primes with level >= 3
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The smallest solutions of ''q''_''p''(''a'') ≡ 0 (mod ''p'') with ''a'' = ''n'' are:

:2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... {{OEIS, id=A039951

A pair (''p'', ''r'') of prime numbers such that ''q''_''p''(''r'') ≡ 0 (mod ''p'') and ''q''_''r''(''p'') ≡ 0 (mod ''r'') is called a Wieferich pair.

References

External links

* Gottfried Helms
Fermat-/Euler-quotients (''a''^''p''-1 – 1)/''p''^''k'' with arbitrary ''k''

* Richard Fischer
Fermat quotients B^(P-1)

1 (mod P^2)

Number theory