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  2. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_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 ...

  3. Discrete logarithm - Wikipedia

    en.wikipedia.org/wiki/Discrete_logarithm

    In number theory, the more commonly used term is index: we can write x = ind r a (mod m) (read "the index of a to the base r modulo m") for r x ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general.

  4. Multiplicative group of integers modulo n - Wikipedia

    en.wikipedia.org/wiki/Multiplicative_group_of...

    The generating set is also chosen to be as short as possible, and for n with primitive root, ... 25 C 20: 20: 20: 2 57 C 2 ×C 18: 36: 18: 2, 20 89 C 88: 88: 88: 3 121

  5. Carmichael function - Wikipedia

    en.wikipedia.org/wiki/Carmichael_function

    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.

  6. Artin's conjecture on primitive roots - Wikipedia

    en.wikipedia.org/wiki/Artin's_conjecture_on...

    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.

  7. Primitive root - Wikipedia

    en.wikipedia.org/wiki/Primitive_root

    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 ...

  8. Dirichlet character - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_character

    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]

  9. Ramanujan's sum - Wikipedia

    en.wikipedia.org/wiki/Ramanujan's_sum

    Let =. Then ζ q is a root of the equation x q − 1 = 0.Each of its powers, ,, …,, = = is also a root. Therefore, since there are q of them, they are all of the roots. The numbers where 1 ≤ n ≤ q are called the q-th roots of unity.