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Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a prime number, then for any integer a, Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats last theorem (edition of 1670). ...
In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
 This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by a, and then subtract a from the resulting number, the final result is divisible by p (see modular arithmetic). Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...
A variant of this theorem is stated in the following form: if p is a prime and a is an integer coprime to p, then Coprime - Wikipedia /**/ @import /skins-1. ...
 If p is a prime number and a is any integer that does not have p as a factor, then a^(p-1) is equal to 1 mod p. In other words, a^(p-1) will always have a remainder of 1 when divided by p.
Fermat's little theorem is the basis for the Fermat primality test. The Fermat primality test is a probabilistic test to determine if a number is composite or probably prime. ...
Numerical examples
Examples of the theorem include: - 43 − 4 = 60 is divisible by 3.
- 72 − 7 = 42 is divisible by 2.
- (−3)7 − (−3) = −2184 is divisible by 7.
- 297 − 2 = 158456325028528675187087900670 is divisible by 97.
History Pierre de Fermat first stated the theorem in a letter dated October 18, 1640 to his friend and confidant Frénicle de Bessy as the following [1]: p divides  whenever p is prime and a is coprime to p. Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ...
October 18 is the 291st day of the year (292nd in leap years). ...
Events December 1 - Portugal regains its independence from Spain and João IV of Portugal becomes king. ...
Bernard Frénicle de Bessy (1605 - 1675) was a French mathematician. ...
Coprime - Wikipedia /**/ @import /skins-1. ...
As usual, Fermat did not prove his assertion, only stating:
- Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.
- (And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid that it would be too long.)
Euler first published a proof in 1736 in a paper entitled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.
The term "Fermat's Little Theorem" was first used in 1913 in Zahlentheorie by Kurt Hensel:
- Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist."
- (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.)
It was first used in English in an article by Irving Kaplansky, "Lucas's Tests for Mersenne Numbers," American Mathematical Monthly, 52 (Apr., 1945).
Further history Chinese mathematicians independently made the related hypothesis (sometimes called the Chinese Hypothesis) that p is a prime if and only if  . It is true that if p is prime, then (this is a special case of Fermat's little theorem). However, the converse (if  then p is prime), and therefore the hypothesis as a whole, is false (e.g. 341=11×31 is a pseudoprime, see below). A mathematician is a person whose area of study and research is mathematics. ...
A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. ...
It is widely stated that the Chinese hypothesis was developed about 2000 years before Fermat's work in the 1600's. Despite the fact that the hypothesis is partially incorrect, it is noteworthy that it may have been known to ancient mathematicians. Some, however, claim that the widely propagated belief that the hypothesis was around so early sprouted from a misunderstanding, and that it was actually developed in 1872. For more on this, see (Ribenboim, 1995).
Proofs Fermat explained his theorem without a proof. The first one who gave a proof was Gottfried Wilhelm Leibniz in a manuscript without a date, where he wrote also that he knew a proof before 1683. Gottfried Wilhelm Leibniz (also von Leibni(t)z)[1] (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath of Sorbian origin, deemed a universal [1] genius in his day and since. ...
Events June 6 - The Ashmolean Museum opens as the worlds first university museum. ...
See Proofs of Fermat's little theorem. This article collects together a variety of proofs of Fermats little theorem, which states that for every prime number p and every integer a (see modular arithmetic). ...
Generalizations A slight generalization of the theorem, which immediately follows from it, is as follows: if p is prime and m and n are positive integers with  , then  In this form, the theorem is used to justify the RSA public key encryption method. In cryptography, RSA is an algorithm for public-key encryption. ...
Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have In number theory, Eulers theorem (also known as the Fermat-Euler theorem or Eulers totient theorem) states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) is Eulers totient function and mod denotes the congruence...
Coprime - Wikipedia /**/ @import /skins-1. ...
 where φ(n) denotes Euler's φ function counting the integers between 1 and n that are coprime to n. This is indeed a generalization, because if n = p is a prime number, then φ(p) = p − 1. The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...
This can be further generalized to Carmichael's theorem. In number theory, the Carmichael function of a positive integer , denoted , is defined as the smallest integer such that for every integer that is coprime to . ...
The theorem has a nice generalization also in finite fields. In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...
Pseudoprimes If a and p are coprime numbers such that  is divisible by p, then p need not be prime. If it is not, then p is called a pseudoprime to base a. F. Sarrus in 1820 found 341 = 11×31 as one of the first pseudoprimes, to base 2. A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. ...
1820 was a leap year starting on Saturday (see link for calendar). ...
A number p that is a pseudoprime to base a for every number a coprime to p is called a Carmichael number (e.g. 561 is a Carmichael number). In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn − 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ...
See also In number theory, Eulers theorem (also known as the Fermat-Euler theorem or Eulers totient theorem) states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) is Eulers totient function and mod denotes the congruence...
A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. ...
In cryptography, RSA is an algorithm for public-key encryption. ...
References - Ribenboim, P. (1995). The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. ISBN 0-387-94457-5.
- János Bolyai and the pseudoprimes (in Hungarian)
External links - Fermat's Little Theorem at cut-the-knot
- Euler Function and Theorem at cut-the-knot
- Fermat's Little Theorem and Sophie's Proof
- Text and translation of Fermat's letter to Frenicle
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