On metallic ratio in Zp

Yamaç Akbiyik S., Akbiyik M., YÜCE S.

Mathematical Methods in the Applied Sciences, cilt.42, sa.16, ss.5535-5550, 2019 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Özet
  • Cilt numarası: 42 Sayı: 16
  • Basım Tarihi: 2019
  • Doi Numarası: 10.1002/mma.5490
  • Dergi Adı: Mathematical Methods in the Applied Sciences
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.5535-5550
  • Anahtar Kelimeler: Fibonacci numbers, generalized Gauss reciprocity, golden ratio, k-Fibonacci number, k-Fibonacci quaternion, metallic ratio
  • İstanbul Gelişim Üniversitesi Adresli: Evet

Özet

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.