On metallic ratio in

Abstract

Metallic ratio is a root of the simple quadratic equation x 2 = k x + 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 . We define k ‐Fibonacci and k ‐Lucas numbers in , and we show that metallic ratio can be calculated in if and only if p ≡ ± 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 for the first time. Moreover, we introduce k ‐Fibonacci and k ‐Lucas quaternions with some algebraic properties and some identities for them.