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The Pell sequence (Pn)∞n=0 is the second order linear recurrence defined by Pn = 2Pn-1 + Pn-2 with initial conditions P0 = 0 and P1 = 1. In this paper, we investigate a generalization of the Pell sequence called the k-generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, The generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced. [ABSTRACT FROM AUTHOR]

Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Petho, we find all k-Fibonacci and k-Lucas numbers which are Fermat numbers. Some more general results are given. [ABSTRACT FROM AUTHOR]