Artin's Conjecture

Artin's Conjecture is a hypothesis in number theory suggesting that for any given prime number \( p \), there exist infinitely many prime numbers \( q \) for which a specific mathematical property holds related to how \( p \) behaves modulo \( q \). In particular, it predicts that a certain "density" of primes will exhibit this behavior, reflecting a deep pattern in the distribution of primes concerning primitive roots—numbers that generate all possible remainders when raised to various powers modulo \( q \). While unproven, it guides research into the fundamental structure of prime numbers and their relationships.