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Some Diophantine Equations involving associated Pell numbers and repdigits

Monalisa Mohapatra, Pritam Kumar Bhoi, Gopal Krishna Panda

Abstract

In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated Pell numbers can be represented as the difference of two repdigits. Additionally, we investigate all associated Pell numbers which are concatenation of three repdigits. Our proof utilizes Baker's theory on linear forms in logarithms of algebraic numbers, along with the Baker-Davenport reduction technique. The computations were carried out with the help of a simple computer program in {\it Mathematica}.

Some Diophantine Equations involving associated Pell numbers and repdigits

Abstract

In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated Pell numbers can be represented as the difference of two repdigits. Additionally, we investigate all associated Pell numbers which are concatenation of three repdigits. Our proof utilizes Baker's theory on linear forms in logarithms of algebraic numbers, along with the Baker-Davenport reduction technique. The computations were carried out with the help of a simple computer program in {\it Mathematica}.

Paper Structure

This paper contains 15 sections, 4 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Results
  3. Repdigits as the difference of two associated Pell numbers
  4. Case 1
  5. Case 2
  6. Reducing the upper bound of $n$
  7. associated Pell numbers which are concatenation of three repdigits
  8. Case 1
  9. Case 2
  10. Case 3
  11. Reducing the upper bound on $n$
  12. associated Pell numbers as the difference of two repdigits
  13. Case 1
  14. Case 2
  15. Reducing the upper bound on $k$

Key Result

Theorem 2.1

m2000. Let $\gamma_1,\ldots,\gamma_l$ be positive real numbers in an algebraic number field $\mathbb{L}$ of degree $d_{\mathbb{L}}$ and $b_1, \ldots, b_l$ be nonzero integers. If $\Gamma = \prod\limits_{i=1}^{l} \gamma_{i}^{b_i} -1$ is not zero, then where $D\geq$ max$\{|b_1|,\ldots,|b_l|\}$ and $A_1, \cdots, A_l$ are positive integers such that $A_j \geq h'(\gamma_j)$ = $max\{d_{\mathbb{L}}h(\ga

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof