On some open problems concerning perfect powers
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916105581232128 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10163 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On some open problems concerning perfect powers Ripà, Marco General Mathematics 00A08 (Primary) 11B50, 11-04 (Secondary) 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. |
| title | On some open problems concerning perfect powers |
| topic | General Mathematics 00A08 (Primary) 11B50, 11-04 (Secondary) |
| url | https://arxiv.org/abs/2205.10163 |