On the Diophantine Equation x 2 + 2 α 5 β 13 γ = y n

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].