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Artin's conjecture on primitive roots The (now proved) conjecture that finite fields are quasi-algebraically closed; see Chevalley–Warning theorem The (now disproved) conjecture that any algebraic form over the p-adics of degree d in more than d 2 variables represents zero: that is, that all p -adic fields are C 2 ; see Ax–Kochen theorem or ...
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.
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.
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 ...
For these primes, 2 is a primitive root modulo p, so 2 n modulo p can be any natural number between 1 and p − 1. =. These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of () /.
In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) ().
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic; Primitive nth root of unity amongst the solutions of z n = 1 in a field; See ...
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.