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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.
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 ...
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 ...
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-theory: ⇒Gromov-Lawson ...
Artin's conjecture on primitive roots; B. ... Second Hardy–Littlewood conjecture; Sister Beiter conjecture; T. Twin prime conjecture; W. Waring–Goldbach problem;
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 ...
Even if we knew that 2 and 3 were both primitive roots modulo infinitely many primes (even the "right" density of primes), I don't think there's any way to conclude that 6 is a primitive root modulo infinitely many primes. - Greg Martin —The preceding unsigned comment was added by 154.20.163.96 23:01, 10 May 2007 (UTC).
Artin billiards; Artin braid group; Artin character; Artin conductor; Artin's conjecture for conjectures by Artin. These include; Artin's conjecture on primitive roots; Artin conjecture on L-functions; Artin group; Artin–Hasse exponential; Artin L-function; Artin reciprocity; Artin–Rees lemma; Artin representation; Artin–Schreier theorem ...