Math-4216 · mathematics · Number theory

Are there infinitely many prime triplets

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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.

The two prime-triplet patterns are $(p,p+2,p+6)$ and $(p,p+4,p+6)$, with every entry prime. Let $T_1$ and $T_2$ be the sets of prime starting values $p$ for these respective patterns.

Statement recorded in the supplied source.

Are there infinitely many prime triplets; equivalently, is $T_1\cup T_2$ infinite? [S2]

Short English statement

Are there infinitely many three-prime configurations of diameter six in either of the two triplet patterns?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400159 (AMR-093-0159). 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 prime triplets. 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, A022004: Initial members of prime triples (p, p+2, p+6)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A022/A022004.seq.
S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A022005: Initial members of prime triples (p, p+4, p+6)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A022/A022005.seq.

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 4216
UnsolvedMath: Catalogue entry