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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:
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]
As a consequence of Lagrange's theorem, the order of a (mod n) always divides φ(n). If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.
Primitive root modulo n, in number theory; Primitive element (field theory), an element that generates a given field extension; Primitive element (finite field), an element that generates the multiplicative group of a finite field; Primitive element (lattice), an element in a lattice that is not a positive integer multiple of another element in ...
A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer. If a primitive root modulo m exists, then there are exactly φ ( φ ( m )) such primitive roots, where φ is the Euler's totient function.
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 ...
Wilson's theorem allows one to define the p-adic ... The values of m for which the product is −1 are precisely the ones where there is a primitive root modulo m ...