On the x-coordinates of Pell equations that are sums of two Padovan numbers.

Let ( P n ) n ≥ 0 be the sequence of Padovan numbers defined by P 0 = 0 , P 1 = P 2 = 1 , and P n + 3 = P n + 1 + P n for all n ≥ 0 . In this paper, we find all positive square-free integers d such that the Pell equations x 2 - d y 2 = N with N ∈ { ± 1 , ± 4 } , have at least two positive integer solutions (x, y) and ( x ' , y ' ) such that both x and x ' are sums of two Padovan numbers.