Math-4208 · mathematics · Number theory

Are there infinitely many Kummer primes

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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 the primes, and put $P_n=\prod_{j=1}^n p_j$ for $n\ge1$. A Kummer number is $K_n=P_n-1$, and a Kummer prime is a prime value of this sequence.

Statement recorded in the supplied source.

Are there infinitely many $n\ge1$ for which $K_n$ is prime? [S2]

Short English statement

Are infinitely many primorials minus one prime?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400150 (AMR-093-0150). 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 there infinitely many Kummer primes. 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, A057588: Kummer numbers: -1 + product of first n consecutive primes., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A057/A057588.seq.
S4 Wikipedia contributors, Euclid number, revision 1321323502 (2026) Kummer numbers. https://en.wikipedia.org/w/index.php?title=Euclid+number&oldid=1321323502.

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 4208
UnsolvedMath: Catalogue entry