Solutions of a generalized markoff equation in Fibonacci numbers

Abstract

In this paper, we find all the solutions (X, Y, Z) = (FI, FJ, FK), where FI, FJ, and FK represent nonzero Fibonacci numbers, satisfying a generalization of Markoff equation called the Jin-Schmidt equation: AX2 + BY2 + CZ2 = DXYZ + 1.

Solutions of the Markoff equation in Tribonacci numbers

“…In the same year, Tengely [18] found all of the triples with Fibonacci terms of the so called Markoff-Rosenberger equation. In my recent result with Tengely [5], we studied the solutions, that are presenting Fibonacci numbers, of another generalization called the Jin-Schmidt equation. Furthermore, Szalay, Tengely and I [6] gave general results regarding the Markoff-Rosenberger triples (G i , G j , G k ) with generalized Lucas number components.…”

Bartz-Marlewski equation with generalized Lucas components

“…where (x, y, z) = (F i , F j , F k ) under the condition that i, j, k ≥ 2 and i ≤ j ≤ k. Furthermore, Togbé, Kafle and Srinivasan [4] obtained the triples (P i , P j , P k ) that satisfy equation (3) under the same conditions. Recently with Tengely [2], we investigated the positive solutions with Fibonacci terms for the Jin-Schmidt equation, which is a generalization of equation ( 3). Moreover, with Szalay, Tengely [3] we studied the solutions (x, y, z) = (U i , U j , U k ) or (V i , V j , V k ) with i, j, k ≥ 1 of another generalization of Markoff equation that is called the Markoff-Rosenberger equation.…”