Carlitz’s Equations on Generalized Fibonacci Numbers

Abstract

Carlitz solved some Diophantine equations on Fibonacci or Lucas numbers. We extend his results to the sequence of generalized Fibonacci and Lucas numbers. In this paper, we solve the Diophantine equations of the form An1⋯Ank=Bm1⋯BmrCt1⋯Cts, where (An), (Bm), and (Ct) are generalized Fibonacci or Lucas numbers. Especially, we also find all solutions of symmetric Diophantine equations Ua1Ua2⋯Uam=Ub1Ub2⋯Ubn, where 1<a1≤a2≤⋯≤am, and 1<b1≤b2≤⋯≤bn.