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  2. Canon arithmeticus - Wikipedia

    en.wikipedia.org/wiki/Canon_arithmeticus

    Jacobi's original tables use 10 or –10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses the smallest possible positive primitive root (Fletcher 1958). The term "canon arithmeticus" is occasionally used to mean any table of indices and powers of primitive roots.

  3. 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.

  4. 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 ...

  5. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    The number 3 is a primitive root modulo 7 [5] because = = = = = = = = = = = = (). Here we see that the period of 3 k modulo 7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7.

  6. 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

  7. Wikipedia:Reference desk/Archives/Mathematics/2010 October 27 ...

    en.wikipedia.org/wiki/Wikipedia:Reference_desk/...

    1.1 Primitive roots modulo 5^n. 3 comments. ... 1.3 1 = -1 ? 5 comments. 1.4 how high do cryptographic functions scale (how big primes can you pick?) 5 comments.

  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. Safe and Sophie Germain primes - Wikipedia

    en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes

    Safe primes ending in 7, that is, of the form 10n + 7, are the last terms in such chains when they occur, since 2(10n + 7) + 1 = 20n + 15 is divisible by 5. For a safe prime, every quadratic nonresidue, except -1 (if nonresidue [a]), is a primitive root. It follows that for a safe prime, the least positive primitive root is a prime number. [15]