Fermat's little theorem

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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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^p \equiv a \pmod{p}\,\!}

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).

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

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Fermat's little theorem is the basis for the Fermat primality test.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{p-1}-1\,} whenever p is prime and a is coprime to p.

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"Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^p \equiv 2 \pmod{p}\,} . It is true that if p is prime, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^p \equiv 2 \pmod{p}\,} (this is a special case of Fermat's little theorem). However, the converse (if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,2^p \equiv 2 \pmod{p}} then p is prime), and therefore the hypothesis as a whole, is false (e.g. 341=11×31 is a pseudoprime, see below).

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.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\equiv n\pmod{p-1}\,} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^m\equiv a^n\pmod{p} \quad\forall a\in\mathbb{Z}.} In this form, the theorem is used to justify the RSA public key encryption method.

Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have

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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.

This can be further generalized to Carmichael's theorem.

The theorem has a nice generalization also in finite fields.

Pseudoprimes

If a and p are coprime numbers such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,a^{p-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 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).

See also

References

External links