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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]
Carmichael calls an element a for which () is the least power of a congruent to 1 (mod n) a primitive λ-root modulo n. [3] This is not to be confused with a primitive root modulo n , which Carmichael sometimes refers to as a primitive φ {\displaystyle \varphi } -root modulo n .)
If g is a primitive root modulo p, then g is also a primitive root modulo all powers p k unless g p −1 ≡ 1 (mod p 2); in that case, g + p is. [14] If g is a primitive root modulo p k, then g is also a primitive root modulo all smaller powers of p. If g is a primitive root modulo p k, then either g or g + p k (whichever one is odd) is a ...
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, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) ().
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.
Equivalently, the formula can be derived by the same argument applied to the multiplicative group of the n th roots of unity and the primitive d th roots of unity. The formula can also be derived from elementary arithmetic. [19] For example, let n = 20 and consider the positive fractions up to 1 with denominator 20: