On the size of the intersection of two Lucas sequences of distinct type II
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Abstract
Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togbé and Walsh proved that the Diophantine equation has at most three solutions in positive integers. Moreover, they showed that if max{a, b } > 4.76 · 1051, then there are at most two positive integer solutions (x, k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.
$$x^2 - a\left( {\frac{{b^k - 1}}
{{b - 1}}} \right)^2 = 1$$
Keywords
Diophantine equation exponential equation linear forms in logarithmsMSC(2000)
11D61 11Y50 11J70Preview
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