Abstract
Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations
assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).
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The author expresses his gratitude to the anonymous reviewer for the instructive suggestions.
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Irmak, N. On solutions of the simultaneous Pell equations \( x^{2}-\left( a^{2}-1\right) y^{2}=1\) and \(y^{2}-pz^{2}=1\) . Period Math Hung 73, 130–136 (2016). https://doi.org/10.1007/s10998-016-0137-0
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DOI: https://doi.org/10.1007/s10998-016-0137-0