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Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved. D. R. Heath-Brown proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes p. [3] He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
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.
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 ...
This sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395...% of the primes. Binary full reptend primes
Square-free. Square-free integer; ... Primitive root modulo n. Multiplicative order; ... Goldbach's conjecture. Goldbach's weak conjecture;
Agoh–Giuga conjecture; Agrawal's conjecture; Andrica's conjecture; Artin's conjecture on primitive roots; B. Bateman–Horn conjecture; Brocard's conjecture;
Carmichael calls an element a for which () is the least power of a congruent to 1 (mod n) a primitive λ-root modulo n. [3] This is not to be confused with a primitive root modulo n , which Carmichael sometimes refers to as a primitive φ {\displaystyle \varphi } -root modulo n .)
1. Euler's theorem can be proven using concepts from the theory of groups: [3] The residue classes modulo n that are coprime to n form a group under multiplication (see the article Multiplicative group of integers modulo n for details).