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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.
The Canon arithmeticus is a set of mathematical tables of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by Carl Gustav Jacob Jacobi . The tables were at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables, they have now been replaced by ...
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.
Toggle the table of contents. Primitive root. ... Print/export Download as PDF; ... In mathematics, a primitive root may mean: Primitive root modulo n in ...
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) (). If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. [1]
Download as PDF; Printable version; In other projects ... −1 is a primitive root mod 4 ... there is a one-to-one correspondence between ...
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 .)