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On some open problems concerning perfect powers

Marco Ripà

Abstract

The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a general theorem that rules out $4/9$ of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence $A352991$ (which includes all the terms of $A001292$). Our risky conjecture states that all the perfect powers belonging to the sequence $A352991$ are perfect squares and they cannot be written as higher order perfect powers if the given term of $A352991$ is not equal to one. This challenging conjecture has been checked for any integer smaller than $10111121314151617181920212223456789$ and no counterexample has been found so far.

On some open problems concerning perfect powers

Abstract

The starting point of our paper is Kashihara's open problem number , concerning the sequence of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the -th term and provide a general theorem that rules out of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence (which includes all the terms of ). Our risky conjecture states that all the perfect powers belonging to the sequence are perfect squares and they cannot be written as higher order perfect powers if the given term of is not equal to one. This challenging conjecture has been checked for any integer smaller than and no counterexample has been found so far.
Paper Structure (5 sections, 3 theorems, 2 equations)

This paper contains 5 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. The exclusion criterion
  3. Perfect cubes in A353025
  4. Conclusion
  5. Appendix

Key Result

Theorem 2.1

For any $m > 1$, $A352991(n)$ cannot be a perfect power of an integer if $A352991(n)$ is a permutation of $A007908(m)$ and $m : m \equiv \{2, 3, 5, 6\} \pmod 9$.

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.1
  • ...and 2 more