9-modularity and GCD properties of generalized Fibonacci numbers (Q2926282)

For a natural number \(n \geq 3\), let \(G_{n}(a, b) = G_{n-1}(a, b) + G_{n-2}(a, b)\), with \(G_{1}(a, b) = a\), \(G_{2}(a, b) = b\), where \(a\), \(b \in \mathbb{Z}\). The sequence \(\{G_{n}(a, b)\}\) is called generalized Fibonacci sequence. In this paper the authors study what conditions on \(n\) and \(a\), \(b\) are needed to ensure that \(G_{n}(a, b)\) could be divisible by a prime power. They prove necessary and sufficient conditions depending on \(n\), \(a\), \(b\) to determine whether \(G_{n}(a, b)\) is divisible by \(9\). The proofs are based on the properties of congruences and the regular Fibonacci numbers. They give a description of the conditions by constructing Latin squares and quasigroups. In the last section of the paper the authors investigate what conditions are needed to guarantee that the greatest common divisor of two generalized Fibonacci number is a regular Fibonacci number. They prove necessary and sufficient conditions by using elementary tools.