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In [[number theory]], the '''Fermat quotient''' of an integer ''a'' with respect to an odd [[prime number|prime]] ''p'' is defined as:<ref>{{MathWorld|title = Fermat Quotient|urlname = FermatQuotient}}</ref><ref name=pg>[http://primes.utm.edu/glossary/xpage/FermatQuotient.html Fermat Quotient] at ''The Prime Glossary''</ref><ref>[[Paulo Ribenboim]], ''13 Lectures on Fermat's Last Theorem'' (1979), especially pp. 152, 159-161.</ref><ref>[[Paulo Ribenboim]], ''My Numbers, My Friends: Popular Lectures on Number Theory'' (2000), p. 216.</ref>
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:<math>q_p(a) = \frac{a^{p-1}-1}{p}.</math>
 
or
 
:<math> \delta_p(a) = \frac{a - a^p }{p} </math>.
 
This article is about the former. For the later see [[p-Derivation]]
 
If the base ''a'' is [[coprime]] to the exponent ''p'' then [[Fermat's little theorem]] says that ''q''<sub>''p''</sub>(''a'') will be an integer. The quotient is named after [[Pierre de Fermat]].
 
==Properties==
From the definition, it is obvious that
:<math>q_p(1) \equiv 0 \pmod{p}</math>
:<math>q_p(-a) \equiv q_p(a) \pmod{p}</math>, since ''p'' &minus; 1 is even.
 
In 1850 [[Gotthold Eisenstein]] proved that if ''a'' and ''b'' are both coprime to ''p'', then:<ref>[[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</ref>
 
:<math>q_p(ab)\equiv q_p(a)+q_p(b) \pmod{p}</math>;
:<math>q_p(a^r)\equiv rq_p(a) \pmod{p}</math>;
:<math>q_p(p-a)\equiv q_p(a) + \frac{1}{a} \pmod{p}</math>;
:<math>q_p(p+a)\equiv q_p(a) - \frac{1}{a} \pmod{p}</math>;
:<math>q_p(p-1)\equiv 1 \pmod{p} </math>;
:<math>q_p(p+1)\equiv -1 \pmod{p}</math>.
 
Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply
 
:<math>q_p(1/a) \equiv -q_p(a) \pmod{p}</math>;
:<math>q_p(a/b) \equiv q_p(a) - q_p(b) \pmod{p}</math>.
 
In 1895 [[Dmitry Mirimanoff]] pointed out that an iteration of Eisenstein's rules gives the corollary:<ref>[[Dmitry Mirimanoff]], "Sur la congruence (''r''<sup>''p'' &minus; 1</sup> &minus; 1):''p'' = ''q<sub>''r''</sub>'' (mod ''p'')," ''Journal für die reine und angewandte Mathematik'' '''115''' (1895): 295-300</ref>
 
:<math>q_p(a+np)\equiv q_p(a)-n\cdot\frac{1}{a} \pmod{p}.</math>
From this, it follows<ref>[[Paul Bachmann]], ''Niedere Zahlentheorie'', 2 vols. (Leipzig, 1902), 1:159.</ref> that
 
:<math>q_p(a+np^2)\equiv q_p(a) \pmod{p}.</math>
 
== Special Values==
 
Eisenstein discovered that the Fermat quotient with base 2 could be expressed as the sum of the reciprocals of the numbers lying in the first half of the range {1, ''p'' &minus; 1}:
 
:<math>-2q_p(2) \equiv \sum_{k=1}^{\frac{p-1}{2}} \frac{1}{k} \pmod{p}.</math>
 
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:
 
:<math>-3q_p(2) \equiv \sum_{k=1}^{\lfloor\frac{p}{4}\rfloor} \frac{1}{k} \pmod{p}.</math><ref>[[James Whitbread Lee Glaisher]], "On the Residues of ''r''<sup>''p'' &minus; 1</sup> to Modulus ''p''<sup>2</sup>, ''p''<sup>3</sup>, etc.," ''Quarterly Journal of Pure and Applied Mathematics'' '''32''' (1901): 1-27.</ref>
 
:<math>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}.</math><ref>Ladislav Skula, "A note on some relations among special sums of reciprocals modulo ''p''," ''Mathematica Slovaca'' '''58''' (2008): 5-10.</ref>
 
:<math>2q_p(2) \equiv \sum_{k=\lfloor\frac{p}{6}\rfloor+1}^{\lfloor\frac{p}{3}\rfloor} \frac{1}{k} \pmod{p}.</math><ref>Emma Lehmer, "On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson," ''Annals of Mathematics'' '''39''' (1938): 350–360, pp. 356ff.</ref> <ref>Karl Dilcher and Ladislav Skula, "A New Criterion for the First Case of Fermat's Last Theorem," ''Mathematics of Computation'' '''64''' (1995): 363-392.</ref>
 
Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:
 
:<math>-3q_p(3) \equiv 2\sum_{k=1}^{\lfloor\frac{p}{3}\rfloor} \frac{1}{k} \pmod{p}.</math><ref>[[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.</ref>
 
:<math>-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}.</math><ref>[[Mathias Lerch]], "Zur Theorie des Fermatschen Quotienten…," ''Mathematische Annalen'' '''60''' (1905): 471-490.</ref>
 
==Generalized Wieferich primes==
If ''q''<sub>''p''</sub>(''a'') ≡ 0 (mod ''p'') then ''a''<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>). Primes for which this is true for ''a'' = 2 are called [[Wieferich prime]]s. In general they are called ''Wieferich primes base a.'' Known solutions of ''q''<sub>''p''</sub>(''a'') ≡ 0 (mod ''p'') for small prime values of ''a'' are:<ref name=pg/>
 
:{| class="wikitable"
|-----
! ''a''
! ''p''
! [[OEIS]] sequence
|-----
| 2 || 1093, 3511
| {{OEIS link|id=A001220}}
|-----
| 3 || 11, 1006003
| {{OEIS link|id=A014127}}
|-----
| 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
| {{OEIS link|id=A123692}}
|-----
| 7 || 5, 491531
| {{OEIS link|id=A123693}}
|-----
| 11 || 71
|
|-----
| 13 || 2, 863, 1747591
| {{OEIS link|id=A128667}}
|-----
| 17 || 2, 3, 46021, 48947, 478225523351
| {{OEIS link|id=A128668}}
|-----
| 19 || 3, 7, 13, 43, 137, 63061489
| {{OEIS link|id=A090968}}
|-----
| 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
| {{OEIS link|id=A128669}}
|}
The smallest solutions of ''q''<sub>''p''</sub>(''a'') ≡ 0 (mod ''p'') with ''a'' = the ''n''th prime  are"
:1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5, … {{OEIS|id=A174422}}.
 
A pair (''p'',''r'') of prime numbers such that ''q''<sub>''p''</sub>(''r'') ≡ 0 (mod ''p'') and ''q''<sub>''r''</sub>(''p'') ≡ 0 (mod ''r'') is called a [[Wieferich pair]].
 
==References==
<references/>
 
==External links==
* Gottfried Helms. [http://go.helms-net.de/math/expdioph/fermatquotients.pdf Fermat-/Euler-quotients (''a''<sup>''p''-1</sup> – 1)/''p''<sup>''k''</sup> with arbitrary ''k''].
* Richard Fischer. [http://www.fermatquotient.com/FermatQuotienten/ Fermat quotients B^(P-1) == 1 (mod P^2)].
 
[[Category:Number theory]]

Latest revision as of 10:07, 17 May 2014

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