The Diophantine equation \(T_l=\mathcal {U}_n -\mathcal {U}_m\)

Abstract

In this paper, we have determined the necessary and sufficient conditions so that Tribonacci numbers can be written as the differences of two elements of generalized Lucas sequences. We have also shown that the number of solutions of the equation in the title is finite. For application, we have determined the Tribonacci numbers written as the difference of two Fibonacci numbers.