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On metallic ratio in Z p
Author(s) -
Yamaç Akbiyik Seda,
Akbiyik Mücahit,
Yüce Salim
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5490
Subject(s) - fibonacci number , mathematics , golden ratio , lucas number , integer (computer science) , combinatorics , generalization , quadratic equation , mathematical analysis , pure mathematics , geometry , computer science , programming language
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.