Mathematics Colloquium

It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed X^{^2}+Y^{^4} is prime infinitely often, and Heath-Brown showed the same for X^{^3}+2Y^{^3}. We will describe recent work which gives a family of multivariate sparse polynomials all of which take infinitely many prime values.