On the Divisibilities \(P_{k}\mid P^{2}_{x}+P_{x}+1\) , \(Q_{k}\mid P^{2}_{x}+P_{x}+1\) , and \(P_{k}\mid Q^{2}_{x}+Q_{x}+1\)

Let \( P_{n} \) , and \( Q_{n} \) be the \(n\mathrm{{th}}\) Pell and Pell-Lucas numbers, respectively. In this paper, we prove that \(P_{k}\) never divides neither \(P^{2}_{x}+P_{x}+1\) nor \(Q^{2}_{x}+Q_{x}+1\) , for \(k\ge 2\) and that \(Q_{k}\) never divides \(P^{2}_{x}+P_{x}+1\) , for any nonnegative integer k and for any nonnegative integer x.