Fermat quotient

In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as^[1][2][3][4]

${\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},}$

or

${\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}}$.

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

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

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]

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

Lerch's formula

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.

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 {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]

Generalized Wieferich primes

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]

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.

References

1. Weisstein, Eric W. "Fermat Quotient". MathWorld.
2. "The Prime Glossary: Fermat quotient". t5k.org. Retrieved 2024-03-16.
3. Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (1979), especially pp. 152, 159-161.
4. Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (2000), p. 216.
5. Gotthold Eisenstein, "Neue Gattung zahlentheoret. Funktionen, die v. 2 Elementen abhangen und durch gewisse lineare Funktional-Gleichungen definirt werden," Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Königl. Preuß. Akademie der Wissenschaften zu Berlin 1850, 36-42
6. Dmitry Mirimanoff, "Sur la congruence (r^{p − 1} − 1):p = q_r (mod p)," Journal für die reine und angewandte Mathematik 115 (1895): 295-300
7. Paul Bachmann, Niedere Zahlentheorie, 2 vols. (Leipzig, 1902), 1:159.
8. Lerch, Mathias (1905). "Zur Theorie des Fermatschen Quotienten ${\displaystyle {\frac {a^{p-1}-1}{p}}=q(a)}$". Mathematische Annalen. 60: 471–490. doi:10.1007/bf01561092. hdl:10338.dmlcz/120531. S2CID 123353041.
9. Sondow, Jonathan (2014). "Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771". arXiv:1110.3113 [math.NT].
10. Sondow, Jonathan; MacMillan, Kieren (2011). "Reducing the Erdős-Moser equation ${\displaystyle 1^{n}+2^{n}+\cdots +k^{n}=(k+1)^{n}}$ modulo ${\displaystyle k}$ and ${\displaystyle k^{2}}$". arXiv:1011.2154 [math.NT].
11. James Whitbread Lee Glaisher, "On the Residues of r^{p − 1} to Modulus p², p³, etc.," Quarterly Journal of Pure and Applied Mathematics 32 (1901): 1-27.
12. Ladislav Skula, "A note on some relations among special sums of reciprocals modulo p," Mathematica Slovaca 58 (2008): 5-10.
13. Emma Lehmer, "On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson," Annals of Mathematics 39 (1938): 350–360, pp. 356ff.
14. Karl Dilcher and Ladislav Skula, "A New Criterion for the First Case of Fermat's Last Theorem," Mathematics of Computation 64 (1995): 363-392.
15. James Whitbread Lee Glaisher, "A General Congruence Theorem relating to the Bernoullian Function," Proceedings of the London Mathematical Society 33 (1900-1901): 27-56, at pp. 49-50.
16. Mathias Lerch, "Zur Theorie des Fermatschen Quotienten…," Mathematische Annalen 60 (1905): 471-490.
17. Wieferich primes to bases up to 1052
18. "Wieferich.txt primes to bases up to 10125". Archived from the original on 2014-07-29. Retrieved 2014-07-22.
19. Wieferich prime in prime bases up to 1000 Archived 2014-08-09 at the Wayback Machine
20. Wieferich primes with level >= 3

External links

• Gottfried Helms. Fermat-/Euler-quotients (ap-1 – 1)/pk with arbitrary k.
• Richard Fischer. Fermat quotients B^(P-1) == 1 (mod P^2).