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On the Diophantine equation \(\binom{n}{k} = \binom{m}{l} + d\). (English) Zbl 1451.11020

In this paper, the solutions of the Diophantine equation \(\binom{n}{k}=\binom{m}{l} + d\), in positive integers \(n\) and \(m\) are studied. First, suppose that \(p\) is a prime \(> 4\) \((k, l) = (2, 4)\), \(d\in \mathbb{Z}\), 3 is a quadratic non-residue modulo \(p\), and the \(p\)-adic valuation of \(12d + 1\) is odd. Then, it is proved that the congruence \(\binom{n}{k}\equiv\binom{m}{l}+d \pmod p\) has no solution, and so the above Diophantine equation has no solution for \((k, l) = (2, 4)\), and \(d \in \mathbb{Z}\). Next, using elementary methods, all integral solutions of the Diophantine equation, with \(l = k \in \{3, 4, 5\}\), and \(d \in\{1, 2, \ldots , 20\}\) are computed. The cases where \( d \in \{-3, \ldots , 3\}\) and \(n \geq k\), \(m \geq l\) are reduced to the computation of integral points of some elliptic curves, and so, all solutions of the above equation are computed. Finally, all integral solutions of the equation with \(d \in \{-3,\ldots, 3\}\), \(k = 2\), \(l = 5\) are computed, by reduction of this problem to the same problem for some curves of genus 2.

MSC:

11D25 Cubic and quartic Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G05 Elliptic curves over global fields
11J86 Linear forms in logarithms; Baker’s method

Software:

WIN4; Magma; SageMath
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Full Text: DOI arXiv

References:

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