The Lebesgue-Nagell-Ramanujan Equation

                               Samir Siksek
                               Oman/Warwick

The Lebesgue-Nagell-Ramanujan equation is the equation $x^2+D=y^n$
in integer unknowns $x$, $y$ and unknown exponent $n \ge 3$. This
equation has been studied since 1850. By 1993 it has been solved
by ad hoc methods for all but $19$ values of the range

$1 \leq D \leq 100$. In this talk we explain how the equation has
recently been solved for all $D$ in the range $1 \leq D \leq 100$,
using a combination of tools from Diophantine approximation and
from Wiles' proof of Fermat's Last Theorem. This talk is based on 
joint work with Y. Bugeaud and M. Mignotte.