On Concatenations of Fibonacci and Lucas Numbers

ALAN M.

BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY, cilt.48, sa.5, ss.2725-2741, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 48 Sayı: 5
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1007/s41980-021-00668-7
  • Dergi Adı: BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.2725-2741
  • Anahtar Kelimeler: Fibonacci Numbers, Lucas Numbers, Diophantine equations, Linear forms in logarithms
  • Yıldız Teknik Üniversitesi Adresli: Evet

Özet

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.