Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
Mathematica Slovaca, cilt.74, sa.3, ss.563-576, 2024 (SCI-Expanded, Scopus)
- 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.