On the diophantine equation x2 + 2α5β13γ = yn

Authors:
Edray Goins;Florian Luca;Alain Togbé
Affiliations:
Department of Mathematics, Purdue University, West Lafayette, IN;Instituto de Matemáticas UNAM, Morelia, Michoacán, Mexico;Department of Mathematics, Purdue University North Central, Westville, IN
Venue:
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Year:
2008

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The MAGMA algebra system I: the user language

Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference

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Abstract

In this paper, we find all the solutions of the Diophantineequation x2 + 2α 5β13γ = yn in nonnegative integers x, y, α, β, γ, n ≥ 3with x and y coprime. In fact, for n = 3, 4, 6, 8, 12, we transform the aboveequation into several elliptic equations written in cubic or quartic modelsfor which we determine all their {2, 5, 13}-integer points. For n ≥ 5, weapply a method that uses primitive divisors of Lucas sequences. Againwe are able to obtain several elliptic equations written in cubic modelsfor which we find all their {2, 5, 13}-integer points. All the computationsare done with MAGMA [12].