Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers

Altassan A., ALAN M.

Mathematica Slovaca, cilt.74, sa.3, ss.563-576, 2024 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 74 Sayı: 3
  • Basım Tarihi: 2024
  • Doi Numarası: 10.1515/ms-2024-0042
  • Dergi Adı: Mathematica Slovaca
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.563-576
  • Anahtar Kelimeler: 11J86, Primary 11B39
  • Yıldız Teknik Üniversitesi Adresli: Evet

Özet

Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10d Fm + Lk and Fn = 10d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.