The $X$-coordinates of Pell equations and Padovan numbers

Authors: SALAH EDDINE RIHANE, MOHAND OUAMAR HERNANE, ALAIN TOGBE

Abstract: In this paper, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=k$, where $k\in\{\pm1,\pm4\}$, which is a Padovan number, with a few exceptions that we completely characterize.

Keywords: Padovan numbers, Pell equation, Linear form in logarithms, reduction method

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