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Integers representable as differences of linear recurrence sequences.

Robert Tichy1, Ingrid Vukusic2, Daodao Yang1

  • 1Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria.

Research in Number Theory
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

This study analyzes integers formed by differences between two linear recurrence sequences. It finds that the density of such integers within a given range is zero, meaning they are very sparse.

Keywords:
Diophantine equationsPillai’s problemRecurrence sequence

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Area of Science:

  • Number Theory
  • Analytic Number Theory
  • Diophantine Equations

Background:

  • Linear recurrence sequences are fundamental in number theory.
  • Understanding the distribution of values generated by sequences is a key problem.
  • Representing integers as differences of sequence terms has implications in various mathematical fields.

Purpose of the Study:

  • To determine the asymptotic behavior of integers representable as differences between terms of two linear recurrence sequences.
  • To quantify the density of such integers within a specified range.
  • To provide a precise formula for counting these integers.

Main Methods:

  • Utilizing tools from analytic number theory.
  • Establishing an asymptotic formula for the count of integers.
  • Analyzing the properties of linear recurrence sequences.

Main Results:

  • An asymptotic formula is derived for the number of integers of the form U_n - V_m within the range [-x, x].
  • The study proves that the density of such integers is zero.
  • This implies that integers representable as differences of terms from two linear recurrence sequences are exceptionally rare.

Conclusions:

  • The set of integers that can be expressed as the difference between terms of two linear recurrence sequences has a density of zero.
  • The established asymptotic formula provides a quantitative measure of this rarity.
  • This result contributes to our understanding of the additive properties of linear recurrence sequences.