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Equal values of pyramidal numbers. (English) Zbl 1446.11051

Summary: Let \(\mathrm{Pyr}_m(x)\) denote the pyramidal number with integer parameters \(m \geq 3\) and \(x \geq 3\). In this note we investigate the Diophantine equation \(\mathrm{Pyr}_m(x) = \mathrm{Pyr}_n(y)\) in positive integer unknowns \(x\) and \(y\), where \(m\) and \(n\) are given, different integers. We deduce an effective upper bound for the size of the solutions. Our proof is based on the basic properties of elliptic curves, and elliptic integrals.

MSC:

11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields

Software:

SageMath; Magma
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References:

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