Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
Number Theory
2025-09-25 v3
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, -adic valuation, Thue equations, and the theory of elliptic curves.
Keywords
Cite
@article{arxiv.2507.16828,
title = {Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors},
author = {Jialai She},
journal= {arXiv preprint arXiv:2507.16828},
year = {2025}
}