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Prime k-tuple

A fixed finite offset set considered through common integer translates whose entries are all prime at an occurrence.

Version
v1 · 2026-10-07 · History
Domain-specific #
13988
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Analytic Number Theory, Prime Numbers → Mathematics
Aliases
Prime k-tuple pattern

Core Idea

A prime k-tuple fixes a finite set H of distinct integer offsets and asks whether one common shift n makes every number in n+H prime. The selected Prime quadruplet is one four-place member. An occurrence at one n is a finite fact; whether an admissible H recurs infinitely often is a separate conjectural question.[ref-23642e0e57d2][ref-15d4d21e9359]

Scope of Application

The entry covers fixed-offset prime configurations in the integers, including pairs and quadruplets. H is admissible when it omits a residue class modulo every prime. That removes a local obstruction to arbitrarily large all-prime translates, but it does not prove one at a chosen n or infinitely many. Small exceptional occurrences can exist even for nonadmissible H: {0,2,4} at n = 3 gives 3,5,7.[^ref-15d4d21e9359]

Clarity

Ask three separate questions: what offsets are fixed; whether a particular n gives only primes; and whether H passes the local residue test. For admissible H = {0,2}, n = 11 succeeds with 11,13, whereas n = 12 fails with 12,14. A successful finite test never by itself proves recurrence.[^ref-15d4d21e9359]

Manages Complexity

The common n binds several primality checks into one configuration. Looking at the offsets modulo p can reveal a fixed-prime-divisor obstruction. If every residue appears, only possible small exceptions remain; if a residue is missing for every p, this specific obstruction is absent. The latter is a diagnostic, not an infinitude proof.[ref-23642e0e57d2][ref-15d4d21e9359]

Abstract Reasoning

The four roles are the integer/primality carrier, a fixed finite set H, one shared shift n, and an all-prime test on n+H. Listing H in another order does not change it. The strict Set and Membership constituent follows from H, while the prime predicate and common translation make this a narrower number-theoretic object.[ref-23642e0e57d2][ref-15d4d21e9359]

Knowledge Transfer

The pair and quadruplet use the same reasoning: specify H, translate it by one n, check each value, then separately inspect local residues. Their exact gaps and residue images differ. H = {0,6}, n = 5 yields the ordinary six-gap prime pair 5,11 under the same general definition.[ref-23642e0e57d2][ref-15d4d21e9359]

Example

Two-place case: H = {0,2} on the positive-integer carrier, with n = 11, yields {11,13}; both are prime. The fixed set, common shift, and all-prime roles are filled. H is admissible, but this one occurrence proves no infinite recurrence.[ref-23642e0e57d2][ref-15d4d21e9359]

Selected four-place case: H = {0,2,6,8} on the same carrier, with n = 11, yields {11,13,17,19}; all four are prime. Its offset set has different size and local residue image. The selected Prime quadruplet is this full-role instance, not an alias for all prime k-tuples.[ref-23642e0e57d2][ref-15d4d21e9359]

Relationships to Other Abstractions

Local relationship map for Prime k-tupleParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Prime k-tupleDOMAINPrime abstraction: Set and Membership — is part ofSet andMembershipPRIME

Current abstraction Prime k-tuple Domain-specific

Parents (1) — more general patterns this builds on

  • Prime k-tuple is part of Set and Membership Prime

    Every admitted configuration contains a fixed finite offset set H with decidable membership.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Prime k-tuple sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Number-Theoretic Properties & Tests (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

The prime k-tuples conjecture concerns infinitely many translates of each admissible H. Hardy and Littlewood’s Theorem X1 is conditional on Hypothesis X; Maynard’s positive-proportion Theorem 1.2 has its own sufficiently large finite-set setup and does not prove the conjecture for every fixed H. The sources do not support a present-day global status claim. The live Pattern Prime has broader roles than one finite arithmetic occurrence.[ref-23642e0e57d2][ref-15d4d21e9359]

References

[^ref-23642e0e57d2]: Hardy, G. H., and J. E. Littlewood (1923). Some Problems of Partitio Numerorum; III: On the Expression of a Number as a Sum of Primes. Acta Mathematica 44, 1–70. DOI: 10.1007/BF02403921. Full original scan: §5.2 printed p.42; §5.62 p.54; §5.66 pp.60–61; §5.68 p.63. Theorem X1 is conditional on Hypothesis X; formula OCR is unreliable.

[^ref-15d4d21e9359]: Maynard, James (2015). Small gaps between primes. Annals of Mathematics 181(1), 383–413. DOI: 10.4007/annals.2015.181.1.7. Full publisher PDF: §1 printed p.383 defines admissibility and states the conjecture; Theorem 1.2 is on printed p.385 and has a sufficiently large finite-set setup. Its publication-time status sentence is not a present-day global status claim.