On the Diophantine equation $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{m}{l}\right)+d$

Abstract

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{m}{l}\right)+d$ for $-3\le d\le 3$ and $(k,l)\in \{(2,3),(2,4),(2,5),(2,6),(2,8),(3,4),(3,6),(4,6),(4,8)\}.$ Moreover, we present some other observations of computational and theoretical nature concerning the title equation.

References in corpus (3)

• Two-cover descent on hyperelliptic curves
• An explicit theory of heights for hyperelliptic Jacobians of genus three
• Diophantine equations coming from binomial near-collisions