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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 g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
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:
For example, if =, every positive integer less than 17 is a 16th root of unity modulo 17, and the integers that are primitive 16th roots of unity modulo 17 are exactly those such that / (). Finding a primitive k th root of unity modulo n
The other four elements of the multiplicative group modulo 15, namely 1, 4 (which satisfies ), 11, and 14, are not primitive λ-roots modulo 15. For a contrasting example, if n = 9, then () = = and (()) =. There are two primitive λ-roots modulo 9, namely 2 and 5, each of which is congruent to the fifth power of the other. They are also both ...
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF( q ) is called a primitive element if it is a primitive ( q − 1) th root of unity in GF( q ) ; this means that each non-zero element of GF( q ) can be written as α i for some natural number i .
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic; Primitive nth root of unity amongst the solutions of z n = 1 in a field; See ...
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.