In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, the Fermat quotient of an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''a'' with respect to an
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' is defined as
[Fermat Quotient]
at ''The Prime Glossary''
:
or
:
.
This article is about the former; for the latter see
''p''-derivation. The quotient is named after
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
.
If the base ''a'' is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to the exponent ''p'' then
Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
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
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
:142857 × 1 = 142857
:14 ...
, and ''p'' will be a
full reptend prime
In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat q ...
.
Properties
From the definition, it is obvious that
:
In 1850,
Gotthold Eisenstein
Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician. He specialized in number theory and mathematical analysis, analysis, and proved several results that eluded even Carl Friedrich Gauss, Gauss. Like ...
proved that if ''a'' and ''b'' are both coprime to ''p'', then:
:
Eisenstein likened the first two of these
congruences
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
to properties of
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s. These properties imply
:
In 1895,
Dmitry Mirimanoff
Dmitry Semionovitch Mirimanoff (russian: Дми́трий Семёнович Мирима́нов; 13 September 1861, Pereslavl-Zalessky, Russia – 5 January 1945, Geneva, Switzerland) became a doctor of mathematical sciences in 1900, in Gen ...
pointed out that an iteration of Eisenstein's rules gives the
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
:
:
From this, it follows that:
:
Lerch's formula
M. Lerch proved in 1905 that
:
Here
is the
Wilson quotient The Wilson quotient ''W''(''p'') is defined as:
:W(p) = \frac
If ''p'' is a prime number, the quotient is an integer by Wilson's theorem; moreover, if ''p'' is composite, the quotient is not an integer. If ''p'' divides ''W''(''p''), it is called ...
.
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 :
:
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:
:
:
:
Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:
:
:
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 prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...
s. 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]
/ref>
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