On metallic ratio in Zp

Metallic ratio is a root of the simple quadratic equation x2  =  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 Zp. We define k‐Fibonacci and k‐Lucas numbers in Zp, and w...

Full description

Saved in:
Bibliographic Details
Published in:Mathematical methods in the applied sciences 2019-11, Vol.42 (16), p.5535-5550
Main Authors: Yamaç Akbiyik, Seda, Akbiyik, Mücahit, Yüce, Salim
Format: Article
Language:English
Subjects:
Eigenvalues
Eigenvectors
Fibonacci numbers
generalized Gauss reciprocity
golden ratio
Integers
k‐Fibonacci number
k‐Fibonacci quaternion
Mathematical analysis
metallic ratio
Numbers
Quadratic equations
Quaternions
Reciprocity
Reciprocity theorem
Silver
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Metallic ratio is a root of the simple quadratic equation x2  =  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 Zp. We define k‐Fibonacci and k‐Lucas numbers in Zp, and we show that metallic ratio can be calculated in Zp if and only if p≡ ± 1  mod (k2 + 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 Z79 for the first time. Moreover, we introduce k‐Fibonacci and k‐Lucas quaternions with some algebraic properties and some identities for them.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.5490