Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Narayana numbers which are products of four $b$-repdigits with a consequence

Passimzouwé Dagou, Pagdame Tiebekabe, Kokou Tcharie

TL;DR

The paper characterizes Narayana numbers that are products of four repdigits in base $g$ for $2\le g\le 12$. It combines Baker-type lower bounds for nonzero linear forms in logarithms with a Dujella–Pethő reduction to obtain extremely large initial exponent bounds, then successively tightens these bounds and conducts a computational search. The main achievement is a complete, explicit determination of all such Narayana numbers within the stated base range, along with finite, provable bounds on the exponents involved. The methodology extends prior work on repdigit products and demonstrates the effectiveness of combining analytic bounds with reduction and computation in multi-exponent Diophantine problems.

Abstract

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers. Moreover we have fully determined them.

On Narayana numbers which are products of four $b$-repdigits with a consequence

TL;DR

The paper characterizes Narayana numbers that are products of four repdigits in base for . It combines Baker-type lower bounds for nonzero linear forms in logarithms with a Dujella–Pethő reduction to obtain extremely large initial exponent bounds, then successively tightens these bounds and conducts a computational search. The main achievement is a complete, explicit determination of all such Narayana numbers within the stated base range, along with finite, provable bounds on the exponents involved. The methodology extends prior work on repdigit products and demonstrates the effectiveness of combining analytic bounds with reduction and computation in multi-exponent Diophantine problems.

Abstract

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base , where is an integer with . We prove that for between and , there are finitely many of these numbers. Moreover we have fully determined them.
Paper Structure (9 sections, 7 theorems, 124 equations, 5 tables)

This paper contains 9 sections, 7 theorems, 124 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Preliminaries
  4. Some properties of Narayana sequence
  5. Linear form in logarithms
  6. Reduction method
  7. Proofs of main result
  8. Proof of Theorem \ref{['theorm1']}
  9. Proof of Theorem \ref{['theorem2']}

Key Result

Theorem 1

Let $g\geq 2$ be an integer. Then the Diophantine equation has only finitely many solutions in integers $k, d_1, d_2, d_3, d_4, \ell, m, n, t$ such as Furthermore, we have In the following theorem, we completely and explicitly give all solutions of equation (eq1) corresponding to $2\leq g\leq 12$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Definition 3.1
  • Lemma 1
  • proof
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 1 more