On metallic ratio in Zp

Abstract

Metallic ratio is a root of the simple quadratic equation x(2) = kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k-Fibonacci (k-Lucas) numbers. This paper is about the metallic ratio in Z(p). We define k-Fibonacci and k-Lucas numbers in Z(p), and we show that metallic ratio can be calculated in Z(p) if and only if p equivalent to +/- 1 mod (k(2) + 4), which is the generalization of the Gauss reciprocity theorem for any integer k. Also, we obtain that the golden ratio, the silver ratio, and the bronze ratio, the three together, can be calculated in Z(79) for the first time. Moreover, we introduce k-Fibonacci and k-Lucas quaternions with some algebraic properties and some identities for them.