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

Abstract

AbstractLet $$ (P_{n})_{n\ge 0} $$ ( P n ) n ≥ 0 be the sequence of Padovan numbers defined by $$ P_0=0 $$ P 0 = 0 , $$ P_1 = P_2=1$$ P 1 = P 2 = 1 , and $$ P_{n+3}= P_{n+1} +P_n$$ P n + 3 = P n + 1 + P n for all $$ n\ge 0 $$ n ≥ 0 . In this paper, we find all positive square-free integers d such that the Pell equations $$ x^2-dy^2 = N $$ x 2 - d y 2 = N with $$ N\in \{\pm 1, \pm 4\} $$ N ∈ { ± 1 , ± 4 } , have at least two positive integer solutions (x, y) and $$(x^{\prime }, y^{\prime })$$ ( x ′ , y ′ ) such that both x and $$x^{\prime }$$ x ′ are sums of two Padovan numbers.