Fermat quotient

From the definition, it is obvious that

${\displaystyle {\begin{aligned}q_{p}(1)&\equiv 0&&{\pmod {p}}\\q_{p}(-a)&\equiv q_{p}(a)&&{\pmod {p}}\quad ({\text{since }}2\mid p-1)\end{aligned}}}$

In 1850, Gotthold Eisenstein proved that if a and b are both coprime to p, then:[5]

${\displaystyle {\begin{aligned}q_{p}(ab)&\equiv q_{p}(a)+q_{p}(b)&&{\pmod {p}}\\q_{p}(a^{r})&\equiv rq_{p}(a)&&{\pmod {p}}\\q_{p}(p\mp a)&\equiv q_{p}(a)\pm {\tfrac {1}{a}}&&{\pmod {p}}\\q_{p}(p\mp 1)&\equiv \pm 1&&{\pmod {p}}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}q_{p}\!\left({\tfrac {1}{a}}\right)&\equiv -q_{p}(a)&&{\pmod {p}}\\q_{p}\!\left({\tfrac {a}{b}}\right)&\equiv q_{p}(a)-q_{p}(b)&&{\pmod {p}}\end{aligned}}}$

In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:[6]

${\displaystyle q_{p}(a+np)\equiv q_{p}(a)-n\cdot {\tfrac {1}{a}}{\pmod {p}}.}$

From this, it follows that:[7]

${\displaystyle q_{p}(a+np^{2})\equiv q_{p}(a){\pmod {p}}.}$

M. Lerch proved in 1905 that[8][9][10]

${\displaystyle \sum _{j=1}^{p-1}q_{p}(j)\equiv W_{p}{\pmod {p}}.}$

Here ${\displaystyle W_{p}}$ is the Wilson quotient.

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 {1, ..., p − 1}:

${\displaystyle -2q_{p}(2)\equiv \sum _{k=1}^{\frac {p-1}{2}}{\frac {1}{k}}{\pmod {p}}.}$

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:

${\displaystyle -3q_{p}(2)\equiv \sum _{k=1}^{\lfloor {\frac {p}{4}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}$[11]

${\displaystyle 4q_{p}(2)\equiv \sum _{k=\lfloor {\frac {p}{10}}\rfloor +1}^{\lfloor {\frac {2p}{10}}\rfloor }{\frac {1}{k}}+\sum _{k=\lfloor {\frac {3p}{10}}\rfloor +1}^{\lfloor {\frac {4p}{10}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}$[12]

${\displaystyle 2q_{p}(2)\equiv \sum _{k=\lfloor {\frac {p}{6}}\rfloor +1}^{\lfloor {\frac {p}{3}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}$[13][14]

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

${\displaystyle -3q_{p}(3)\equiv 2\sum _{k=1}^{\lfloor {\frac {p}{3}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}$[15]

${\displaystyle -5q_{p}(5)\equiv 4\sum _{k=1}^{\lfloor {\frac {p}{5}}\rfloor }{\frac {1}{k}}+2\sum _{k=\lfloor {\frac {p}{5}}\rfloor +1}^{\lfloor {\frac {2p}{5}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}$[16]

If q_p(a) ≡ 0 (mod p) then a^p−1 ≡ 1 (mod p²). 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:[2]

a	p (checked up to 5 × 1013)	OEIS sequence
1	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)	A000040
2	1093, 3511	A001220
3	11, 1006003	A014127
4	1093, 3511
5	2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801	A123692
6	66161, 534851, 3152573	A212583
7	5, 491531	A123693
8	3, 1093, 3511
9	2, 11, 1006003
10	3, 487, 56598313	A045616
11	71
12	2693, 123653	A111027
13	2, 863, 1747591	A128667
14	29, 353, 7596952219	A234810
15	29131, 119327070011	A242741
16	1093, 3511
17	2, 3, 46021, 48947, 478225523351	A128668
18	5, 7, 37, 331, 33923, 1284043	A244260
19	3, 7, 13, 43, 137, 63061489	A090968
20	281, 46457, 9377747, 122959073	A242982
21	2
22	13, 673, 1595813, 492366587, 9809862296159	A298951
23	13, 2481757, 13703077, 15546404183, 2549536629329	A128669
24	5, 25633
25	2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
26	3, 5, 71, 486999673, 6695256707
27	11, 1006003
28	3, 19, 23
29	2
30	7, 160541, 94727075783

For more information, see [17][18][19] and.[20]

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, ... (sequence A039951 in the OEIS)

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.

• 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).