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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. So g is a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n.
For a primitive () th root x, the number () / is a primitive th root of unity. If k does not divide λ ( n ) {\displaystyle \lambda (n)} , then there will be no k th roots of unity, at all. Finding multiple primitive k th roots 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:
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.
For n = 1, the cyclotomic polynomial is Φ 1 (x) = x − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive n th root of unity for every n > 1. As Φ 2 (x) = x + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive n th root of unity for every even n > 2.
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 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]
There are four primitive λ-roots modulo 15, namely 2, 7, 8, and 13 as . The roots 2 and 8 are congruent to powers of each other and the roots 7 and 13 are congruent to powers of each other, but neither 7 nor 13 is congruent to a power of 2 or 8 and vice versa.