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  2. Artin's conjecture on primitive roots - Wikipedia

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

    Let a be an integer that is not a square number and not −1. Write a = a 0 b 2 with a 0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then the conjecture states S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.

  3. Root of unity modulo n - Wikipedia

    en.wikipedia.org/wiki/Root_of_unity_modulo_n

    For a primitive () th root x, the number () / is a primitive th root of unity. If k does not divide λ ( n ) {\displaystyle \lambda (n)} , then there will be no k th roots of unity, at all. Finding multiple primitive k th roots modulo n

  4. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    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.

  5. Dirichlet character - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_character

    If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: χ 16 , 9 {\displaystyle \chi _{16,9}} is induced from χ 8 , 5 {\displaystyle \chi _{8,5}} and χ 16 , 15 {\displaystyle \chi _{16,15}} and χ 8 , 7 {\displaystyle \chi _{8,7 ...

  6. Root of unity - Wikipedia

    en.wikipedia.org/wiki/Root_of_unity

    For n = 1, the cyclotomic polynomial is Φ 1 (x) = x − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive n th root of unity for every n > 1. As Φ 2 (x) = x + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive n th root of unity for every even n > 2.

  7. Carmichael function - Wikipedia

    en.wikipedia.org/wiki/Carmichael_function

    For example, if n = 15, then λ(n) = 4 while () = and (()) =. 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.

  8. Multiplicative group of integers modulo n - Wikipedia

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

    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:

  9. Primitive polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Primitive_polynomial...

    The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to 3 2 − 1 = 8 are 1, 3, 5, and 7, the four primitive roots in GF(3 2) are α, α 3 = 2α + 1, α 5 = 2α, and α 7 = α + 2. The primitive roots α and α 3 are algebraically