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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 ...
Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group (Z/nZ)*. Such a generator has order |(Z/nZ)*| = n−1 and both equivalences will hold for any such primitive root. Note that if there exists an a < n such that the first equivalence fails, a is called a Fermat witness for the compositeness of n.
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.
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
The Lucas test relies on the fact that the multiplicative order of a number a modulo n is n − 1 for a prime n when a is a primitive root modulo n. If we can show a is primitive for n, we can show n is prime.
One such subgroup P, is the set of diagonal matrices [], x is any primitive root of F q. Since the order of F q is q − 1, its primitive roots have order q − 1, which implies that x (q − 1)/p n or x m and all its powers have an order which is a power of p.
Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m, there is an integer k such that g k ≡ a (mod m). A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer.
The first paragraph after the sub-head "Table of primitive roots" explains that this is not a table of smallest primitive roots; it is Gauss's table of primitive roots, which are chosen to given 10 the smallest index. So 6 is chosen as the listed primitive root for 13 because 6 2 = 10 mod 13, whereas 2 10 = 10 mod 13. I agree that placing the ...