Fermat's little theorem states that if  is a prime number, then for any integer , the number  is an integer multiple of .
In the notation of modular arithmetic, this is expressed as
a^p \equiv a \pmod p.
For example, if  = 2 and  = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7.
If  is not divisible by , Fermat's little theorem is equivalent to the statement that  is an integer multiple of , or in symbols:..
a^{p-1} \equiv 1 \pmod p.
For example, if  = 2 and  = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory.
The theorem is named after Pierre de Fermat, who stated it in 1640.
It is called the "little theorem" to distinguish it from Fermat's Last Theorem..
History
thumb|right|Pierre de Fermat Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy.
His formulation is equivalent to the following:
If  is a prime and  is any integer not divisible by , then  is divisible by .
Fermat's original statement was
This may be translated, with explanations and formulas added in brackets for easier understanding, as:
Every prime number [] divides necessarily one of the powers minus one of any [geometric] progression []  [that is, there exists  such that  divides ], and the exponent of this power [] divides the given prime minus one [divides ].
After one has found the first power [] that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question [that is, all multiples of the first  have the same property].
Fermat did not consider the case where  is a multiple of  nor prove his assertion, only stating: (in French)
(And this proposition is generally true for all series [sic] and for all prime numbers; I would send you a demonstration of it, if I did not fear going on for too long.) for the English translation
Euler provided the first published proof in 1736, in a paper titled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the Proceedings of the St. Petersburg Academy, but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.
The term "Fermat's little theorem" was probably first used in print in 1913 in Zahlentheorie by Kurt Hensel:
(There is a fundamental theorem holding in every finite group, usually called Fermat's little theorem because Fermat was the first to have proved a very special part of it.)
An early use in English occurs in A.A. Albert's Modern Higher Algebra (1937), which refers to "the so-called 'little' Fermat theorem" on page 206.
Further history
Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese Hypothesis) that  if and only if  is prime.
Indeed, the "if" part is true, and it is a special case of Fermat's little theorem.
However, the "only if" part is false: For example, , but 341 = 11 × 31 is a pseudoprime.
See below.
Proofs
Several proofs of Fermat's little theorem are known.
It is frequently proved as a corollary of Euler's theorem.
Generalizations
Euler's theorem is a generalization of Fermat's little theorem: for any modulus  and any integer  coprime to , one has
a^{\varphi (n)} \equiv 1 \pmod n,
where  denotes Euler's totient function (which counts the integers from 1 to  that are coprime to ).
Fermat's little theorem is indeed a special case, because if  is a prime number, then .
A corollary of Euler's theorem is: for every positive integer , if the integer  is coprime with  then
x \equiv y \pmod{\varphi(n)}\quad\text{implies}\quad a^x \equiv a^y \pmod n,
for any integers  and .
This follows from Euler's theorem, since, if x \equiv y \pmod{\varphi(n)}, then  for some integer , and one has
a^x = a^{y + \varphi(n)k} =  a^y (a^{\varphi(n)})^k \equiv a^y 1^k \equiv a^y \pmod n.
If  is prime, this is also a corollary of Fermat's little theorem.
This is widely used in modular arithmetic, because this allows reducing modular exponentiation with large exponents to exponents smaller than .
Euler's theorem is used with  not prime in public-key cryptography, specifically in the RSA cryptosystem, typically in the following way: if
y=x^e\pmod n,
retrieving  from the values of ,  and  is easy if one knows .If  is not coprime with , Euler's theorem does not work, but this case is sufficiently rare for not being considered.
In fact, if it occurred by chance, this would provide an easy factorization of , and thus break the considered instance of RSA.
In fact, the extended Euclidean algorithm allows computing the modular inverse of  modulo , that is the integer  such that ef\equiv 1\pmod{\varphi(n)}.
It follows that
x\equiv x^{ef}\equiv (x^e)^f \equiv y^f  \pmod n.
On the other hand, if  is the product of two distinct prime numbers, then .
In this case, finding  from  and  is as difficult as computing   (this has not been proven, but no algorithm is known for computing  without knowing ).
Knowing only , the computation of  has essentially the same difficulty as the factorization of , since , and conversely, the factors  and  are the (integer) solutions of the equation .
The basic idea of RSA cryptosystem is thus: if a message  is encrypted as , using public values of  and , then, with the current knowledge, it cannot be decrypted without finding the (secret) factors  and  of .
Fermat's little theorem is also related to the Carmichael function and Carmichael's theorem, as well as to Lagrange's theorem in group theory.
Converse
The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers.
However, a slightly stronger form of the theorem is true, and it is known as Lehmer's theorem.
The theorem is as follows:
If there exists an integer  such that
a^{p-1}\equiv 1\pmod p
and for all primes  dividing  one has
a^{(p-1)/q}\not\equiv 1\pmod p,
then  is prime.
This theorem forms the basis for the Lucas primality test, an important primality test, and Pratt's primality certificate.
Pseudoprimes
If  and  are coprime numbers such that  is divisible by , then  need not be prime.
If it is not, then  is called a (Fermat) pseudoprime to base .
The first pseudoprime to base 2 was found in 1820 by Pierre Frédéric Sarrus: 341 = 11 × 31.
A number  that is a Fermat pseudoprime to base  for every number  coprime to  is called a Carmichael number (e.g. 561).
Alternately, any number  satisfying the equality
\gcd\left(p, \sum_{a=1}^{p-1} a^{p-1}\right)=1
is either a prime or a Carmichael number.
Miller–Rabin primality test
The Miller–Rabin primality test uses the following extension of Fermat's little theorem:
If  is an odd prime, and  with  odd, then for every   prime to , either , or there exists  such that  and .
This result may be deduced from Fermat's little theorem by the fact that, if  is an odd prime, then the integers modulo  form a finite field, in which 1 has exactly two square roots, 1 and −1.
The Miller–Rabin test uses this property in the following way: given , with  odd, an odd integer for which primality has to be tested, choose randomly  such that ; then compute ; if  is not 1 nor −1, then square it repeatedly modulo  until you get 1, −1, or have squared  times.
If  and −1 has not been obtained, then   is not prime.
Otherwise,  may be prime or not.
If  is not prime, the probability that this is proved by the test is higher than 1/4.
Therefore, after   non-conclusive random tests, the probability that  is not prime is lower than , and may thus be made as low as desired, by increasing  .
In summary, the test either proves that a number is not prime, or asserts that it is prime with a probability of error that may be chosen as low as desired.
The test is very simple to implement and computationally more efficient than all known deterministic tests.
Therefore, it is generally used before starting a proof of primality.
See also
Fermat quotient
Frobenius endomorphism
-derivation
Fractions with prime denominators: numbers with behavior relating to Fermat's little theorem
RSA
Table of congruences
Modular multiplicative inverse
Notes
References
Further reading
Paulo Ribenboim (1995).
The New Book of Prime Number Records (3rd ed.).
New York: Springer-Verlag. .
pp.
22–25, 49.
External links
János Bolyai and the pseudoprimes (in Hungarian)
Fermat's Little Theorem at cut-the-knot
Euler Function and Theorem at cut-the-knot
Fermat's Little Theorem and Sophie's Proof
