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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 ...
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. ... Artin's conjecture on primitive roots; B. Bateman–Horn ...
10 Andrews–Curtis conjecture: combinatorial group theory: James J. Andrews and Morton L. Curtis: 358 Andrica's conjecture: 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'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 ...
Also, the current version states that it suffices to prove Artin's conjecture for prime numbers a; I don't believe this is correct. 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 ...
He left two conjectures, both known as Artin's conjecture. The first concerns Artin L-functions for a linear representation of a Galois group ; and the second the frequency with which a given integer a is a primitive root modulo primes p , when a is fixed and p varies.