I will report on joint work with Yann Bugeaud, Maurice Mignotte, Samir Siksek and Szabolcs Tengely on a new method to determine all integral solutions to an equation of the form $y2 = f(x)$. One ingredient is an astronomical, but reasonable and computable bound on~$|x|$, the other is a sieving procedure using the group of rational points on the Jacobian variety of the curve given by the equation. I will illustrate this approach by solving the binomial coefficient equation $\binom{y}{2}=\binom{x}{5}$.