Event Details

A discussion of an ongoing research topic, intended to be accessible to current students. Speaker: Ashleigh Ratcliffe (Leicester)

Title: Solving Diophantine equations: A systematic approach

Abstract: Diophantus of Alexandria, who lived in the Third Century, was interested in determining integer solutions to polynomial equations with integer coefficients. Such equations are now known as (polynomial) Diophantine equations. In 1970, Yuri Matiyasevich proved the negative answer to Hilbert’s tenth problem, showing that there is no general method that can solve all Diophantine equations. Instead, researchers focus on solving restricted classes of equations. This is an active area of research, but, traditionally, it is a matter of researcher’s taste which equations to solve.

This talk reports on the following project: define a suitable notion of "size'' of a polynomial Diophantine equation, order all equations by size, and solve them in this order. On the way, we discuss some methods for solving certain classes of Diophantine equations, develop a general method to solve all two-variable three-monomial equations, and deduce the complete list of integers |n| ≤ 200 which are representable as the difference of two fourth powers.