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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:
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:
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. The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility ...
As the 3rd and the 7th roots of unity belong to GF(4) and GF(8), respectively, the 54 generators are primitive n th roots of unity for some n in {9, 21, 63}. Euler's totient function shows that there are 6 primitive 9 th roots of unity, 12 primitive 21 st roots of unity, and 36 primitive 63 rd roots of unity.
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]
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at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables they have now been replaced by ...
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.