On Tribonacci numbers written as a product of two Perrin numbers

Abstract

In this paper, we give all solutions of the Diophantine equation Tn = RkRm, where (n, k, m) ∈ Z+ x Z+ x Z+ , Rk is the Perrin sequence, and Tn is the Tribonacci sequence. We show that this Diophantine equation has only 7 integer solution triples. For the proof, we use Baker’s method. Our motivation is to show that linear forms in logarithms can still be effectively used for the solutions of different Diophantine equations involving classical number sequences such as Fibonacci or Lucas sequences.