Artin’s primitive root conjecture

Establish that for every integer a that is neither a perfect square nor −1, there exist infinitely many primes p such that a is a primitive root modulo p; equivalently, determine whether the set of primes for which a generates the multiplicative group Z_p^* is infinite.

Background

The binary case of completely unclustered BWTs reduces to the existence of infinitely many primes p = 2n+1 for which 2 is a primitive root, directly connecting the combinatorial problem to Artin’s conjecture.

To highlight this connection, the authors restate the conjecture and show, for general k, that the word α^n (where α lists Σ_k in decreasing order) is a BWT image if and only if kn+1 is prime and k is a primitive root modulo kn+1. Thus, the combinatorial question inherits the long-standing status of Artin’s conjecture.