Diophantine equations coming from binomial near-collisions

Abstract

In this paper we solve the Diophantine equation $\left(\genfrac{}{}{0ex}{}{m}{l}\right)-\left(\genfrac{}{}{0ex}{}{n}{k}\right)=d$ (where m,n are positive integers unknowns) when (k,l)=(3,6) for various values of d and when (k,l)=(8,2) and d=1. As a byproduct of our results we will obtain that (k,l)-near collisions with difference 1 do not exist if (k,l)=(6,3), (3,6), (8,2) thus establishing a conjecture stated in the article "Binomial collisions and near collisions" published by A. Blokhuis, A. Brouwer and B. de Weger in Integers 17 (2017), A64.

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Cited by in corpus (1)

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