Abstract
In this paper, we derive some identities on Pell, Pell-Lucas, and balancing numbers and the relationships between them. We also deduce some formulas on the sums, divisibility properties, perfect squares, Pythagorean triples involving these numbers. Moreover, we obtain the set of positive integer solutions of some specific Pell equations in terms of the integer sequences mentioned in the text.
Highlights
Introduction and preliminariesLet p and q be two integers such that d = p2 – 4q = 0
2.1 Sums and divisibility properties In this subsection, we deal with the sums and divisibility properties of numbers mentioned
In 1766, Lagrange proved that the Pell equation x2 – dy2 = 1 has an infinite number of solutions if d is positive and nonsquare
Summary
We derive our main results. 2.1 Sums and divisibility properties In this subsection, we deal with the sums and divisibility properties of numbers mentioned. We reformulate (11) in terms of Pell and Pell-Lucas numbers as follows. The second result can be proved . Theorem 2 For the sequences an, Xn, Yn, Bn, bn, Qn, and Pn, we have: (1) Xn = Pn+1 + Pn for n ≥ 0, and the sum of the first n nonzero terms of Xn is n. Yn = Pn+2 – Pn for n ≥ 0, and the sum of the first n nonzero terms of Yn is n. Since Xn = Pn+1 + Pn and Yn = Pn+2 – Pn, we get Xn + Yn = Pn+1 + Pn+2, and . The remaining cases can be proved . From Theorem 4 we have the following result.
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