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On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

Herbert Batte, Prosper Kaggwa

Abstract

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas number. Mathematically, we study of two Diophantine equations $L_n=b^dL_m+F_k$ and $L_n=b^dF_m+L_k$, where $d$ is the number of digits of $F_k$ or $L_k$ in base $b$. To tackle these equations, we combine tools from Diophantine approximation on non-zero linear forms in logarithms and reduction methods based on continued fractions. This allows us to prove that only finitely many such Lucas numbers exist.

On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

Abstract

Let and denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base mixed concatenations of a Fibonacci number and a Lucas number. Mathematically, we study of two Diophantine equations and , where is the number of digits of or in base . To tackle these equations, we combine tools from Diophantine approximation on non-zero linear forms in logarithms and reduction methods based on continued fractions. This allows us to prove that only finitely many such Lucas numbers exist.
Paper Structure (12 sections, 11 theorems, 133 equations, 2 tables)

This paper contains 12 sections, 11 theorems, 133 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Methods
  5. Some results on Fibonacci and Lucas numbers
  6. Bounds on variables from Equation \ref{['main1']}
  7. Bounds on variables from Equation \ref{['main2']}
  8. Linear Forms in Logarithms
  9. Reduction Methods
  10. Proof of the Main Results
  11. Proof of Theorem \ref{['thm1']}
  12. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.1

For $2\le b\le10$, the only solutions to the Diophantine equation main1 are; In particular, the only Lucas numbers satisfying equation main1 when $b=10$ (decimal base) are

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Logarithmic height
  • Theorem 2.2: Matveev, BugeaudMignotteSiksek2006
  • Lemma 2.3: Dujella & Pethő , duj
  • Lemma 2.4: Legendre, qu
  • Lemma 2.5
  • proof
  • Lemma 2.6: Lemma 7 in guz
  • Lemma 3.1
  • ...and 7 more