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Diophantine equations coming from binomial near-collisions. (English) Zbl 1432.11034

Summary: In this paper, we solve the Diophantine equation \(\binom{m}{l}-\binom{n}{k} = d\) (where \(m\), \(n\) are positive integer unknowns) when \((k, l) = (6,3),(3,6)\) 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 \((k,l) = (3,6),(8,2)\) thus establishing a conjecture stated in the article published in 2017 by A. Blokhuis et al. [Integers 17, Paper A64, 8 p. (2017; Zbl 1414.11026)].

MSC:

11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
11B65 Binomial coefficients; factorials; \(q\)-identities

Citations:

Zbl 1414.11026

Software:

Magma
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Full Text: arXiv Link

References:

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