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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.
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 ...
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.
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
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:
Both 2 and 3 are primitive λ-roots modulo 5 and also primitive roots modulo 5. n = 8. The set of numbers less than and coprime to 8 is {1,3,5,7} . Hence φ(8) = 4 and λ(8) must be a divisor of 4. In fact λ(8) = 2 since ().
Artin's conjecture on primitive roots that if an integer is neither a perfect square nor , then it is a primitive root modulo infinitely many prime numbers Brocard's conjecture : there are always at least 4 {\displaystyle 4} prime numbers between consecutive squares of prime numbers, aside from 2 2 {\displaystyle 2^{2}} and 3 2 {\displaystyle 3 ...
The other primitive q-th roots of unity are the numbers where (a, q) = 1. Therefore, there are φ(q) primitive q-th roots of unity. Thus, the Ramanujan sum c q (n) is the sum of the n-th powers of the primitive q-th roots of unity. It is a fact [3] that the powers of ζ q are precisely the primitive roots for all the divisors of q. Example. Let ...