ABSTRACT
This chapter deals with a prime and looking for a primitive root. Reverse the procedure, starting with an integer and asking for primes for which it is a primitive root. In 1967, Christopher Hooley proved Emil Artin's conjecture on primitive roots, under the assumption of the Generalized Riemann Hypothesis. In 1985, Roger Heath-Brown, inspired by work of Ram Murty and Rajiv Gupta, showed, without any unproved assumptions, that there are at most two primes that are not primitive roots for infinitely many primes. So, for example, at least one of 2, 3, 5 is a primitive root for infinitely many primes, but we don't know whether it's 2, 3, or 5. The chapter explores the asserts that if p is a prime then there exist primitive roots for p. If students know about fields, then they might notice that the proof give also shows that a finite multiplicatively closed subset of a field forms a cyclic group.
