Does every prime number appear in the Euclid–Mullin sequence

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.

Define the Euclid--Mullin sequence recursively by $a_1=2$ and, for $n\ge1$, let $a_{n+1}$ be the least prime divisor of $1+\prod_{j=1}^n a_j$. The product plus one exceeds one, so the next term exists. No prime already in the product can divide this number; thus its terms are pairwise distinct.

Statement recorded in the supplied source.

For every prime $p$, does there exist an index $n\ge1$ with $a_n=p$? [S2]

Short English statement

Does the recursive least-prime-factor Euclid sequence eventually contain every prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400169 (AMR-093-0169). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Andrew R. Booker and Carl Pomerance, Squarefree smooth numbers and Euclidean prime generators, arXiv:1607.01557v4 (2017) Introduction, p. 1. https://arxiv.org/pdf/1607.01557.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Does every prime number appear in the Euclid–Mullin sequence. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A000945: Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of 1 + Product_{k=1..n} a(k)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000945.seq.