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Fermat's little theorem states that if p is a prime number, then for any integer a, In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
This means that if you take some number a, multiply it by itself p times and subtract a, the result is divisible by p (see modular arithmetic). It is often stated in the following equivalent form: if p is a prime and a is an integer coprime to p, then Modular arithmetic is a system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers wrap around after they reach a certain value (the modulus). ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
It is called Fermat's little theorem to differentiate it from Fermat's last theorem. Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. ...
Fermat's little theorem is the basis for the Fermat primality test. The Fermat primality test is a probabalistic test to determine if a number is composite or probably prime. ...
History
Pierre de Fermat found the theorem around 1636. It appeared in one of his letters, dated October 18, 1640 to his confidant Frenicle as the following: p divides ap−1 − 1 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 Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ...
Events February 24 - King Christian of Denmark gives an order that all beggars that are able to work must be sent to Brinholmen Island to build ships or as galley rowers March 26 - Utrecht University founded in The Netherlands. ...
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. ...
Chinese mathematicians independently made the related hypothesis (sometimes called the Chinese Hypothesis) that p is a prime if and only if 2p = 2(mod p). It is true that if p is prime, then 2p = 2(mod p) (this is a special case of Fermat's little theorem). However, the converse (if 2p = 2(mod p) then p is prime), and therefore the hypothesis as a whole, is false (e.g. 341=11×31 is a pseudoprime, see below). 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 Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1, 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...
Events June 6 - The Ashmolean Museum opens as the worlds first university museum. ...
See Proofs of Fermat's little theorem.
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 m ≡ n (mod p − 1), then am ≡ an (mod p) for all integers a. 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) states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 ( mod n) where φ(n) denotes Eulers totient function. ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 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. In number theory, the totient φ(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.
The theorem has a nice generalization also in finite fields. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
Pseudoprimes If a and p are coprime numbers such that ap−1 − 1 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 for all integers b which are relatively prime to n (see modular arithmetic). ...
See also 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. ...
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 (http://bolyai.port5.com/kisfermat.htm) (in Hungarian)
External links - Fermat's Little Theorem (http://www.cut-the-knot.org/blue/Fermat.shtml)
- Euler Function and Theorem (http://www.cut-the-knot.org/blue/Euler.shtml)
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