A Pillai-Catalan-type problem involving Fibonacci numbers


Ibrahimov S. S., Mahmudov N. I.

JOURNAL OF ANALYSIS, vol.32, pp.3039-3046, 2024 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 32
  • Publication Date: 2024
  • Doi Number: 10.1007/s41478-024-00779-4
  • Journal Name: JOURNAL OF ANALYSIS
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Page Numbers: pp.3039-3046
  • Azerbaijan State University of Economics (UNEC) Affiliated: No

Abstract

This paper addresses A Pillai-Catalan-type problem assosiated with Fibonacci numbers. Let Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{n}$$\end{document} be the Fibonacci numbers defined by the recurrence relation F1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{1}=1$$\end{document}, F2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{2}=1$$\end{document} and Fn=Fn-1+Fn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{n}=F_{n-1}+F_{n-2}$$\end{document} for all n >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}. We will find all positive integer solution to the equation 3x-Fn2y=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3<^>{x}-F_{n}2<^>{y}=1$$\end{document} using properties of Fibonacci numbers, linear forms of logarithms and Baker-Davenport reduction method.