TRIBONACCI NUMBERS AS PRODUCTS OF TWO NARAYANA’S NUMBERS

Abstract

Let $(N_n)_{n \ge0}$ be the Narayana’s cows sequence and $(T_n)_{n \ge0}$ the Tribonacci sequence. In this paper, we investigate Diophantine equations involving the Tribonacci sequence and the Narayana sequence. More precisely, we solve the Diophantine equations \[T_k=N_mN_n,\] \[N_k=T_mT_n.\] Using a combination of linear forms in logarithms, technical reduction and computational arguments we prove that these equations admit only finitely many solutions. All solutions are explicitly determined.