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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 ...
Such an element is called a primitive λ-root modulo n. The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. [ 1 ] It is also known as Carmichael's λ function , the reduced totient function , and the least universal exponent function .
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 ...
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
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).
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x 3 ≡ p (mod q) is solvable if and only if ...