Are all Euclid numbers square-free

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.

Let $p_1=2<p_2=3<\cdots$ enumerate primes and $P_n=\prod_{j=1}^{n}p_j$ for $n\ge1$. Put $E_n=P_n+1$, the $n$th Euclid number. A positive integer is squarefree when no prime square divides it.

Statement recorded in the supplied source.

Is $q^2\nmid E_n$ for every $n\ge1$ and every prime $q$? [S2]

Short English statement

Does every primorial plus one have no repeated prime factor?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400138 (AMR-093-0138). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are all Euclid numbers square-free. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A006862: Euclid numbers: 1 + product of the first n primes., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A006/A006862.seq.
• S4 Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv2503.21485v1 (2025) Introduction p1. https://arxiv.org/pdf/2503.21485.
• S5 Wikipedia contributors, Euclid number, revision 1321323502 (2026) Definition and Unsolved problems. https://en.wikipedia.org/w/index.php?title=Euclid+number&oldid=1321323502.