Artin’s conjecture for primitive roots

Abstract

In his preface to Diophantische Approximationen, Hermann Minkowski expressed the conviction that the “deepest interrelationships in analysis are of an arithmetical nature.” Gauss described one such remarkable interrelationship in articles 315–317 of his Disquisitiones Arithmeticae. There, he asked why the decimal fraction of 1/7 has period length 6: $$ x^2 + y^2 = 5 $$ (1) whereas 1/11 has period length of only 2: $$ x - y = 1 $$ (2) Why does 1/99007599, when written as a binary fraction (that is, expanded in base 2), have a period of nearly 50 million 0s and 1s? To answer these questions, Gauss introduced the concept of a primitive root.