On b-concatenations of two k-generalized Fibonacci numbers

ALAN M., Altassan A.

Acta Mathematica Hungarica, 2025 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1007/s10474-025-01517-3
  • Dergi Adı: Acta Mathematica Hungarica
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet
  • Anahtar Kelimeler: concatenation, Diophantine equation, Fibonacci number, k-Fibonacci numbers, k-generalized Fibonacci number, linear form in logarithms
  • Yıldız Teknik Üniversitesi Adresli: Evet

Özet

Let k≥2 be an integer. One of the generalization of the classical Fibonacci sequence is defined by the recurrence relationFn(k)=Fn-1(k)+⋯+Fn-k(k) for all n≥2 with the initial values Fi(k)=0 for i=2-k,…,0 and F1(k)=1.Fn(k) is an order k generalization of the Fibonacci sequence and it is called k-generalizedFibonacci sequence or shortly k-Fibonacci sequence. Banks and Luca [7], among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem in more general manner by taking into account the concatenations of two terms of the same sequence in base b≥2. First, we show that there exists only finitely many such concatenations for each k≥2 and b≥2. Next, we completely determine all these concatenations for all k≥2 and 2≤b≤10.