Abstract
We consider the sequences and which are the generalizations of Fibonacci and Lucas sequences, respectively. Then we determine some identities involving these generalized sequences to present all solutions of the equations and for and a square-free integer . In addition to these, all solutions of some different Diophantine equations such as , , , , , are identified, by using divisibility rules of the sequences and . MSC:11B37, 11B39, 11C20, 11D09, 11D45.
Highlights
In this paper, we consider the generalized Fibonacci sequence and the generalized Lucas sequence
Generalized Fibonacci and Lucas numbers can be extended to negative indices
In the last section, we will find all solutions of Diophantine equations mentioned above
Summary
We consider the generalized Fibonacci sequence (un) and the generalized Lucas sequence (vn). ). Subsequently, if p ≥ and p – is a square-free integer, we will find all solutions of Diophantine equations x – v nxy + y = – p – u n and x – vnxy + y = – p – .
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