Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918146942697472 |
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| author | She, Jialai |
| author_facet | She, Jialai |
| contents | The Erdős-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate potential counterexamples centered around perfect cubes, which are themselves powerful. This paper establishes a new non-existence result for a family of such integer triplets with distinct structural constraints, combining techniques from modular arithmetic, $p$-adic valuation, Thue equations, and the theory of elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors She, Jialai Number Theory The Erdős-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate potential counterexamples centered around perfect cubes, which are themselves powerful. This paper establishes a new non-existence result for a family of such integer triplets with distinct structural constraints, combining techniques from modular arithmetic, $p$-adic valuation, Thue equations, and the theory of elliptic curves. |
| title | Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.16828 |