On metallic ratio in Z p

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 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 ≡ ± 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.