Erdős–Mollin–Walsh conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A positive integer $n$ is powerful when every prime divisor $p\mid n$ satisfies $p^2\mid n$. The integer $1$ is powerful by the empty condition. Equivalently, $n=a^2b^3$ for some positive integer $a$ and squarefree positive integer $b$.

Statement recorded in the supplied source.

Is there no $N\in\mathbb N$ for which $N$, $N+1$ and $N+2$ are all powerful? [S2]

Short English statement

Can three consecutive positive integers ever all have every prime factor appearing to exponent at least two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400122 (AMR-093-0122). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485v1 (2025) Introduction, definition and Conjecture 1, p. 1. https://arxiv.org/pdf/2503.21485.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Erdős–Mollin–Walsh conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.