On Concatenations of Fibonacci and Lucas Numbers

ALAN M.

BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY, vol.48, no.5, pp.2725-2741, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 5
  • Publication Date: 2022
  • Doi Number: 10.1007/s41980-021-00668-7
  • Journal Name: BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.2725-2741
  • Keywords: Fibonacci Numbers, Lucas Numbers, Diophantine equations, Linear forms in logarithms
  • Yıldız Technical University Affiliated: Yes

Abstract

Let (F-n)(n >= 0) and (L-n)(n >= 0) be the Fibonacci and Lucas sequences. In this paper we determine all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. This problem is identical to solve the Diophantine equations F-n = 10(d) L-m + L-k and L-n = 10(d) F-m + F-k in non-negative integers (n, m, k), where d denotes the number of digits of L-k and F-k, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.