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

TL;DR

This paper proves that certain triplets of integers around perfect cubes with prime-square factors cannot exist, advancing understanding of the Erdős-Mollin-Walsh conjecture in number theory.

Contribution

It introduces new non-existence results for specific integer triplets involving perfect cubes and powerful numbers, using advanced number-theoretic techniques.

Findings

No such triplets exist under the given constraints

The methods combine modular arithmetic, $p$-adic valuations, and elliptic curve theory

Results support the conjecture by ruling out potential counterexamples

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.