Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. In this lecture I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis) concerning the bias of cubic Gauss sums over the prime numbers. This explains a well-known numerical bias first observed by Kummer in 1846. This bias was later the subject of testing on some of the first super computers in the 20th century. Time permitting, results on higher order Gauss sums will be discussed.  This is joint work with Maksym Radziwill.