In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined asPaulo Ribenboim, 13 Lectures on Fermat's Last Theorem (1979), especially pp. 152, 159-161.Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (2000), p. 216.
or
.
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 qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
Properties From the definition, it is obvious that
In 1850, Gotthold Eisenstein proved that if a and b are both coprime to p, then: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
Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply
In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:Dmitry Mirimanoff, "Sur la congruence (rp − 1 − 1):p = qr (mod p)," Journal für die reine und angewandte Mathematik 115 (1895): 295-300
From this, it follows that:Paul Bachmann, Niedere Zahlentheorie, 2 vols. (Leipzig, 1902), 1:159.
Lerch's formula
M. Lerch proved in 1905 that
Here 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}:
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:
James Whitbread Lee Glaisher, "On the Residues of rp − 1 to Modulus p2, p3, etc.," Quarterly Journal of Pure and Applied Mathematics 32 (1901): 1-27.
Ladislav Skula, "A note on some relations among special sums of reciprocals modulo p," Mathematica Slovaca 58 (2008): 5-10.
Emma Lehmer, "On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson," Annals of Mathematics 39 (1938): 350–360, pp. 356ff.Karl Dilcher and Ladislav Skula, "A New Criterion for the First Case of Fermat's Last Theorem," Mathematics of Computation 64 (1995): 363-392.
Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:
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.
Mathias Lerch, "Zur Theorie des Fermatschen Quotienten…," Mathematische Annalen 60 (1905): 471-490.
Generalized Wieferich primes If qp(a) ≡ 0 (mod p) then ap−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 qp(a) ≡ 0 (mod p) for small values of a are:
{| class="wikitable"
! a ! p (checked up to 5 × 1013) ! OEIS sequence
| 1 || 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)
| 2 || 1093, 3511
| 3 || 11, 1006003
| 4 || 1093, 3511
| 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
| 6 || 66161, 534851, 3152573
| 7 || 5, 491531
| 8 || 3, 1093, 3511
| 9 || 2, 11, 1006003
| 10 || 3, 487, 56598313
| 11 || 71
| 12 || 2693, 123653
| 13 || 2, 863, 1747591
| 14 || 29, 353, 7596952219
| 15 || 29131, 119327070011
| 16 || 1093, 3511
| 17 || 2, 3, 46021, 48947, 478225523351
| 18 || 5, 7, 37, 331, 33923, 1284043
| 19 || 3, 7, 13, 43, 137, 63061489
| 20 || 281, 46457, 9377747, 122959073
| 21 || 2
| 22 || 13, 673, 1595813, 492366587, 9809862296159
| 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
| 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 Wieferich primes to bases up to 1052Wieferich prime in prime bases up to 1000 and.Wieferich primes with level >= 3
The smallest solutions of qp(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, ...
A pair (p, r) of prime numbers such that qp(r) ≡ 0 (mod p) and qr(p) ≡ 0 (mod r) is called a Wieferich pair.
References
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).