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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.
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.
In analytic number theory and related branches of mathematics, ... For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units ...
The degree of Φ n is given by Euler's totient function, which counts (among other things) the number of primitive n th roots of unity. [9] The roots of Φ n are exactly the primitive n th roots of unity. Galois theory can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals.
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root).
number theory: Dorin Andrica: 45 Artin conjecture (L-functions) number theory: Emil Artin: 650 Artin's conjecture on primitive roots: number theory: ⇐generalized Riemann hypothesis [2] ⇐Selberg conjecture B [3] Emil Artin: 325 Bateman–Horn conjecture: number theory: Paul T. Bateman and Roger Horn: 245 Baum–Connes conjecture: operator K ...
The field () contains a th primitive root of unity if and only if is a divisor of ; if is a divisor of , then the number of primitive th roots of unity in () is () (Euler's totient function). The number of n {\displaystyle n} th roots of unity in G F ( q ) {\displaystyle \mathrm {GF} (q)} is g c d ( n , q − 1 ) {\displaystyle \mathrm {gcd} (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: