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Solutions to a Pillai-type equation involving Tribonacci numbers and S-units

Herbert Batte, Florian Luca

Abstract

Let $ \{T_n\}_{n\geq 0} $ be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation $T_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. In particular, we show that there is no integer $c$ with at least six representations of the form $T_n-2^x3^y$.

Solutions to a Pillai-type equation involving Tribonacci numbers and S-units

Abstract

Let be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation , for . In particular, we show that there is no integer with at least six representations of the form .
Paper Structure (18 sections, 24 theorems, 225 equations)

This paper contains 18 sections, 24 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Methods
  5. Preliminaries
  6. Linear forms in logarithms
  7. Reduced Bases for Lattices and LLL--reduction methods
  8. Equations in $\alpha,\beta,\gamma$
  9. Bounds for solutions to $S-$unit equations
  10. Proof of Theorem \ref{['1.2at']}
  11. Proof of Theorem \ref{['1.2bt']}
  12. An absolute upper bound on $n$
  13. Reduction of the upper bound on $n$
  14. Conclusion
  15. Proof of Theorem \ref{['1.2ct']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

The Diophantine equation 1.2t has in the case that $c = 0$, exactly five solutions, namely Furthermore, this representation is given by

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.1: Lemma 1 in VZ
  • Lemma 2.2: Lemma 7 in GL
  • Definition 2.1
  • Theorem 2.1: Matveev, MAT
  • Definition 2.2
  • ...and 30 more