Local solubility of generalised Fermat equations

Let n > 1. I will discuss how many of the generalised Fermat equations ax^n + by^n + cz^n = 0 are soluble in the real numbers and all p-adic fields as a,b,c varies. The main new analytic tool is a large sieve inequality for power residue symbols. This talk is based on joint work with Peter Koymans, Ross Paterson and Alec Shute.