On the Diophantine equation \(x^2 + 2^{\alpha}5^{\beta}13^{\gamma} = y^n\). (English) Zbl 1232.11130

Goins, Edray; Luca, Florian; Togbé, Alain

van der Poorten, Alfred J. (ed.) et al., Algorithmic number theory. 8th international symposium, ANTS-VIII Banff, Canada, May 17–22, 2008 Proceedings. Berlin: Springer (ISBN 978-3-540-79455-4/pbk). Lecture Notes in Computer Science 5011, 430-442 (2008).

Summary: In this paper, we find all the solutions of the Diophantine equation \[ x^2 + 2^{\alpha} 5^{\beta}13^{\gamma} = y^n \] in nonnegative integers \(x, y, \alpha, \beta, \gamma\), \(n \geq 3\) with \(x\) and \(y\) coprime. In fact, for \(n = 3, 4, 6, 8, 12\), we transform the above equation into several elliptic equations written in cubic or quartic models for which we determine all their \(\{2, 5, 13\}\)-integer points. For \(n\geq 5\), we apply a method that uses primitive divisors of Lucas sequences. Again we are able to obtain several elliptic equations written in cubic models for which we find all their \(\{2, 5, 13\}\)-integer points. All the computations are done with MAGMA.
For the entire collection see [Zbl 1136.11003].

MSC:

11Y50	Computer solution of Diophantine equations
11D61	Exponential Diophantine equations

Keywords:

exponential Diophantine equations; computer solution of Diophantine equations