On some open problems concerning perfect powers

TL;DR

This paper investigates open problems about perfect powers in specific integer sequences, confirming a conjecture up to a large term and proposing a new conjecture about the nature of perfect powers within a sequence, supported by extensive computational checks.

Contribution

It confirms Kashihara's conjecture up to the 100128-th term and introduces a new conjecture that all perfect powers in a certain sequence are perfect squares, supported by large-scale computational verification.

Findings

Confirmed Kashihara's conjecture up to the 100128-th term.

Proposed a new conjecture that all perfect powers in sequence A352991 are perfect squares.

Extensive computational checks found no counterexamples below 10^32.

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.