On a variant of Pillai problem: integers as difference between generalized Pell numbers and perfect powers

The k-generalized Pell sequence P-(k) := (P-n((k)))(n >= -(k-2)) is the linear recurrence sequence of order k, whose first k terms are 0, ..., 0, 1 and satisfies the relation P-n((k)) = 2P(n-1)((k)) + 2P(n-2)((k)) + ... + P-n-k((k)), for all n, k >= 2. In this paper, we investigate about integers that have at least two representations as a difference between a k-Pell number and a perfect power. In order to exhibit a solution method when b is known, we find all the integers c that have at least two representations of the form P-n((k)) - b(m) for b is an element of [2, 10]. This paper extends the previous works in Ddamulira et al. (Proc. Math. Sci. 127: 411-421, 2017) and Erazo et al. (J. Number Theory 203: 294-309, 2019).