Number theory is the study of whole numbers, and ratios of whole numbers (called rational numbers).  In particular, number theorists since antiquity have been interested in finding whole number and rational number solutions to equations, such as y2 = x3 + 22 – i.e., can a square be exactly 22 more than a cube?  The research of my collaborators and myself in recent years has largely concerned the study of the probability that random equations of various types have whole number or rational number solutions.  For example, a recent result of this kind that we proved is that, given a random equation of the form y2 = x3 + a x2 + b x + c, where a, b, c are whole numbers, there is a positive probability that it does not possess any rational number solutions for x and y, and a positive probability that it possesses infinitely many rational solutions.

Awards and Achievements

  • National Academy of Science