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Counting relatively prime pairs of palindromes

Hirotaka Kobayashi, Yuta Suzuki, Ryota Umezawa

Abstract

For a given base $g\ge2$, a positive integer is called a palindrome if its base $g$ expansion reads the same backwards as forwards. In this paper, we give an asymptotic formula for the number of relatively prime pairs of palindromes of a fixed odd length and of any base $g\ge2$, which solves an open problem proposed by Banks and Shparlinski (2005).

Counting relatively prime pairs of palindromes

Abstract

For a given base , a positive integer is called a palindrome if its base expansion reads the same backwards as forwards. In this paper, we give an asymptotic formula for the number of relatively prime pairs of palindromes of a fixed odd length and of any base , which solves an open problem proposed by Banks and Shparlinski (2005).
Paper Structure (4 sections, 10 theorems, 111 equations)

This paper contains 4 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Palindromes in arithmetic progressions
  4. Proof of the main theorems

Key Result

Theorem 1

For $N\ge1$, we have where $c>0$ is some constant, and the constant $c>0$ and the implicit constant depends only on $g$.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4: Banks--Shparlinski BanksShparlinski:PrimeDivisorPalindrome
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7: Tuxanidy--Panario TuxanidyPanario
  • ...and 11 more