On metallic ratio in Zp

Yamaç Akbiyik S., Akbiyik M., YÜCE S.

Mathematical Methods in the Applied Sciences, vol.42, no.16, pp.5535-5550, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Abstract
  • Volume: 42 Issue: 16
  • Publication Date: 2019
  • Doi Number: 10.1002/mma.5490
  • Journal Name: Mathematical Methods in the Applied Sciences
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.5535-5550
  • Keywords: Fibonacci numbers, generalized Gauss reciprocity, golden ratio, k-Fibonacci number, k-Fibonacci quaternion, metallic ratio
  • Istanbul Gelisim University Affiliated: Yes


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.