In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
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 a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
In number theory, Euler's theorem states that, if n and a are coprime positive integers, and is Euler's totient function, then a raised to the power is congruent to 1 modulo n; that is
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
This article collects together a variety of proofs of Fermat's little theorem, which states that
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime.
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is, the factorial satisfies
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.
A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that n is prime.
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840).
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:
In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.
In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin.
In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century.
