On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$

Homero R. Gallegos-Ruiz; Nikolaos Katsipis; Szabolcs Tengely; Maciej Ulas

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arXiv · 1904.11369

On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$

Homero R. Gallegos-Ruiz·Nikolaos Katsipis·Szabolcs Tengely·Maciej Ulas

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Abstract

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ 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.

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Homero R. Gallegos-Ruiz, Nikolaos Katsipis, Szabolcs Tengely, Maciej Ulas. 2019-04-25. On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$. https://arxiv.org/abs/1904.11369

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