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arXiv · 2507.16828

Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors

Jialai She
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Abstract

The Erd\H{o}s-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.

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Jialai She. 2025-07-07. Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors. https://arxiv.org/abs/2507.16828

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