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Primitive root modulo. n. In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which gk ≡ a (mod n). Such a value k is called the index or discrete logarithm ...
Weisstein, Eric W. "Primitive Root". MathWorld. Web-based tool to interactively compute group tables by John Jones; OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic)) Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups:
The n th roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of ...
An nth root of unity is a complex number whose nth power is 1, a root of the polynomial x n − 1. The set of all nth roots of unity forms a cyclic group of order n under multiplication. [1] The generators of this cyclic group are the nth primitive roots of unity; they are the roots of the nth cyclotomic polynomial.
Appearance. In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to ...
where the modulus m is a prime number or a power of a prime number, the multiplier a is an element of high multiplicative order modulo m (e.g., a primitive root modulo n), and the seed X 0 is coprime to m. Other names are multiplicative linear congruential generator (MLCG) [2] and multiplicative congruential generator (MCG).
The above discussion indicates how knowing the factors of n allows us to find the roots efficiently. Say there were an efficient algorithm for finding square roots modulo a composite number. The article congruence of squares discusses how finding two numbers x and y where x 2 ≡ y 2 (mod n) and x ≠ ±y suffices to factorize n efficiently.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ...