Exact divisibility by powers of the Pell and Associated Pell numbers

Let P-n be the n-th Pell number and Q(n) be the n-th Associated Pell number. We obtain certain divisibility and exact divisibility results for the powers of the Pell and Associated Pell numbers by applying the concept of p-adic valuation of the numbers. In particular, we show that P-n(k) parallel to m if and only if P-n(k+1) parallel to P-nm for all m, n >= 2 and k >= 1. In addition, Q(n)(k) parallel to m if and only if Q(n)(k+1) parallel to Q(nm) for all m, n >= 2 and k >= 1 is also proved.