Multiplicative group of integers modulo nIn modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
Fermat numberIn mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
Quadratic residueIn number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
Multiplicative orderIn number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that . In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n. The order of a modulo n is sometimes written as . The powers of 4 modulo 7 are as follows: The smallest positive integer k such that 4k ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3.
Euler's theoremIn number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient 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 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number.
Generating set of a groupIn abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if is a subset of a group , then , the subgroup generated by , is the smallest subgroup of containing every element of , which is equal to the intersection over all subgroups containing the elements of ; equivalently, is the subgroup of all elements of that can be expressed as the finite product of elements in and their inverses.
Wilson's theoremIn 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 (using the notations of modular arithmetic), the factorial satisfies exactly when n is a prime number. In other words, any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n. This theorem was stated by Ibn al-Haytham (c. 1000 AD), and, in the 18th century, by the English mathematician John Wilson.
Discrete logarithmIn mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases.
Primality testA primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input).
Unit (ring theory)In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R^× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).
