# REDIRECT Fermat's little theorem

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.

* Leonhard Euler produces the first published proof of Fermat's " little theorem ".

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

Fermat's little theorem states that if p is prime and a is coprime to p, then a < sup > p − 1 </ sup > − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides a < sup > x − 1 </ sup > − 1, then x is called a Fermat pseudoprime to base a.

It is not a prime, since it equals 11 · 31, but it satisfies Fermat's little theorem: 2 < sup > 340 </ sup > ≡ 1 ( mod 341 ) and thus passes

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a < sup > p </ sup > − a is an integer multiple of p. In the notation of modular arithmetic, this says

If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a < sup > p − 1 </ sup > − 1 is an integer multiple of p:

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory.

It is called the " little theorem " to distinguish it from Fermat's last theorem.

( 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.

This is a special case of Fermat's little theorem.

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

Fermat's little theorem also has a nice generalization in finite fields.

The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers.

* Fractions with prime denominators – numbers with behavior relating to Fermat's little theorem

It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k < sup > q − 2 </ sup > mod q.

Fermat's little theorem.

The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.

This article collects together a variety of proofs of Fermat's little theorem, which states that

Some of the proofs of Fermat's little theorem given below depend on two simplifications.

Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

If the conjecture were true, it would be a generalization of Fermat's last theorem, which could be seen as the special case n = 2: if, then.

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea del Centina, “ she had actually shown that it holds for every exponent p < 197 .” L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

In an unpublished manuscript entitled Remarque sur l ’ impossibilité de satisfaire en nombres entiers a l ’ équation x < sup > p </ sup > + y < sup > p </ sup > = z < sup > p </ sup >, Germain showed that any counterexamples to Fermat's theorem for p > 5 must be numbers “ whose size frightens the imagination ,” around 40 digits long.

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

Fermat's Last Theorem is a particularly well-known example of such a theorem.

On the other hand, Fermat's last theorem has always been known by that name, even before it was proven ; it was never known as " Fermat's conjecture ".

* Persian Muslim astronomer and mathematician, Abu-Mahmud al-Khujandi, invents the astronomical sextant and first states a special case of Fermat's last theorem.

* Pierre de Fermat makes a notation, in a document margin, claiming to have proof of what would become known as Fermat's last theorem.

1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830.

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

The laws of reflection and refraction can be derived from Fermat's principle which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Fermat's principle is the main principle of quantum electrodynamics where it states that any particle ( e. g. a photon or an electron ) propagates over all available ( unobstructed ) paths and the interference ( sum, or superposition ) of its wavefunction over all those paths ( at the point of observer or detector ) gives the correct probability of detection of this particle ( at this point ).

Fermat's principle states that the path light takes between two points is the path that has the minimum optical path length.

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time.

Fermat's little theorem states that if p is prime and < math > 1

Local extrema can be found by Fermat's theorem, which states that they must occur at critical points.

A slightly more rigorous definition of a light ray follows from Fermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

It is trivial to show that p being prime is sufficient for the second equivalence to hold, since if p is prime, Fermat's little theorem states that

It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, scrawled in the margins of Fermat's copy of ' Arithmetica ', which states that the equation, where,, and are non-zero integers, has no solution with greater than 2.

In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as

Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number.

Fermat's theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely ( up to order ), and that no other primes can be represented this way, aside from 2 = 1 < sup > 2 </ sup >+ 1 < sup > 2 </ sup >.