The X -coordinates of Pell equations and Padovan numbers

In this paper, we show that there is at most one value of the positive integer X " role="presentation"> X X X participating in the Pell equation X 2 − d Y 2 = k " role="presentation"> X 2 − d Y 2 = k X 2 − d Y 2 = k X^2-dY^2=k , where k ∈ { ± 1 , ± 4 } " role="presentation"> k ∈ { ± 1 , ± 4 } k ∈ { ± 1 , ± 4 } k\in\{\pm1,\pm4\} , which is a Padovan number, with a few exceptions that we completely characterize.