Artin’s Primitive Root Conjecture represents one of many famous problems in elementary number theory that has resisted complete solution thus far. Significant progress was made in 1967, when Christopher Hooley published a conditional proof of the conjecture under the assumption of a certain case of the Generalised Riemann Hypothesis. In this survey we present a description of the conjecture and the underlying algebraic theory, and provide a detailed account of Hooley’s proof which is intended to be accessible to those with only undergraduate level knowledge. We also discuss a result concerning the qx+1 problem, whose proof requires similar techniques to those used by Hooley.

Author Bio

Shalome Kurian is a fourth-year Mathematics undergraduate at the University of Warwick. He is most interested in pure mathematics and in particular number theory and analysis. During the summer of 2019 he completed a URSS project in which he wrote a survey paper on Artin's Primitive Root Conjecture. In his spare time, he enjoys playing tennis and similar sports.

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