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 here is a fixed finite set H of distinct integer offsets considered with a common integer shift n. An occurrence is a shift for which every number in n+H is prime. The offset set and one all-prime translate are the named formal configuration; the frozen Prime quadruplet is one four-place member, not a synonym for the whole family. Hardy and Littlewood use a common n with fixed distinct offsets, and Maynard states the modern admissibility condition for such sets.^[1]^[2]

The distinction between a set, its finite occurrence, and a conjecture about infinitely many occurrences governs this entry. Admissibility asks whether H omits a residue class modulo every prime. It is a local condition relevant to possible arbitrarily large prime translates. It is not part of the finite occurrence test: H = {0,2,4} is not admissible, yet n = 3 gives the primes 3, 5, 7. Conversely, an admissible H does not make a particular n an occurrence.^[2]

Structural Signature¶

Constitutive roles

Integer and prime-number carrier. Offsets and shifts are integers; the translated values are tested for ordinary primality.
Fixed finite set of distinct offsets. H contains the positions to test; changing H changes the configuration. Listing order and repeated notation do not create new members.
Common integer translate. One n is added to every member of H. Testing different shifts separately is a different question.
All-prime occurrence test. Every entry of n+H must be prime for that n to realize the tuple. A single composite value makes that attempted occurrence fail.

Important but nonconstitutive diagnostics: H may be locally admissible or obstructed, and the number of offsets and their gaps vary. An asymptotic count and infinite recurrence are conjectural or conditional statements about a fixed H, not extra slots in a finite tuple.^[1]^[2]

What It Is Not¶

A prime k-tuple occurrence is not the prime k-tuples conjecture. The latter asserts infinitely many all-prime translates for an admissible H; one checked translate establishes only one occurrence. Hardy and Littlewood’s Theorem X1 is expressly conditional on Hypothesis X. Maynard’s Theorem 1.2 proves a positive-proportion result under its sufficiently large finite-set setup, rather than the conjecture for every fixed H.^[1]^[2]

Nor is an admissible offset set automatically an occurrence. For H = {0,2}, n = 12 gives {12,14}, both composite, even though H passes the local admissibility test. At the other boundary, the exceptional {3,5,7} occurrence shows why nonadmissibility does not exclude every small all-prime translate.^[2]

Scope of Application¶

The scope is arithmetic configurations of prime positions in the integers. It includes two-place patterns such as H = {0,2} and H = {0,6}, and the selected four-place H = {0,2,6,8}; each can be examined at a specified common n. It also supports a separate local-residue diagnostic on H before asking about unbounded recurrence. Hardy and Littlewood compare quadruplet offset forms in §5.68, but their predicted relative frequency and general asymptotic remain conjectural or conditional in the cited original.^[1]

No cross-domain use of “pattern” is needed to recognize the object. The live Prime Pattern has stronger recurrence and evidence roles; one fixed H with one finite prime translate does not inherit that full signature. A finite set of primes can be normalized as n = its least member and H = its differences, so this entry does not invent a mathematical outside case based on a failure to predeclare H.

Clarity¶

Keep three questions separate: What is H? Does this n make every n+h prime? Does H avoid a local obstruction for arbitrarily large translates? They have different answers. For H = {0,2}, n = 11 passes the occurrence test and H is admissible. For the same H, n = 12 fails the occurrence test. For H = {0,2,4}, n = 3 passes the occurrence test even though H occupies every residue modulo 3 and is not admissible.^[2]

Manages Complexity¶

Holding H fixed makes a multi-prime question inspectable. Instead of treating each prime independently, the common n connects all positions, and the residue image of H modulo p detects an obstruction: if every residue appears, every translate has a member divisible by p. This rules out arbitrarily large all-prime translates apart from small cases where that member equals p. It does not supply a general proof of infinitely many translates when no obstruction exists.^[1]^[2]

Abstract Reasoning¶

The elementary model is H = {h₁,…,hₖ} and one integer n, with the predicate “n+h is prime for every h in H.” The map h ↦ n+h preserves differences between positions. For admissibility, reduce H modulo each prime p and ask whether at least one residue class is missing. These operations explain why the two-place and four-place cases share an identity while their residue calculations differ.^[2]

A formal inference must not reverse the local test. If H covers all residues modulo some p, a fixed-prime-divisor obstruction limits large occurrences. If H omits one residue for each p, that removes this specific obstruction; it does not prove prime values at any particular shift or infinitely many such shifts. Hardy and Littlewood’s conditional density prediction and Maynard’s positive-proportion theorem retain their stated hypotheses and quantifiers.^[1]^[2]

Knowledge Transfer¶

A useful transfer within number theory is the workflow “fix offsets → test a common translate → inspect residues → state the recurrence question with its quantifier.” The pair and quadruplet share that workflow; neither inherits the other’s exact gap count or residue image. The six-gap pair H = {0,6}, n = 5 also fits the general fixed-offset definition and yields 5 and 11 without making a two-unit gap constitutive.^[1]^[2]

The abstract set-and-membership constituent can be recognized outside prime arithmetic, but transferring only that set structure does not transfer primality, congruence obstructions, or the Hardy–Littlewood conjecture.

Examples¶

Canonical pair: H = {0,2}, n = 11¶

The carrier is the positive integers with ordinary primality. The fixed offset set has two members, 0 and 2. The common shift 11 yields {11,13}; both entries are prime, so this is an occurrence. H occupies only one class modulo 2 and at most two classes modulo any p ≥ 3, hence it is admissible. This finite check does not prove infinitely many twin-prime occurrences.^[1]^[2]

Mapped back: integer/primality carrier = positive integers; fixed set = {0,2}; common translate = n = 11; all-prime test = 11 and 13 prime.

Selected member: H = {0,2,6,8}, n = 11¶

The carrier and all-prime test are the same kind, but H now has four positions. One shift gives {11,13,17,19}, all prime. The offsets occupy one class modulo 2, two modulo 3, and four modulo 5; for p ≥ 7 four offsets cannot exhaust p classes. Thus this H is admissible, with a local calculation different from the pair. Hardy and Littlewood list the quadruplet form; the displayed n is a finite occurrence, not proof of their predicted asymptotic.^[1]^[2]

Mapped back: integer/primality carrier = positive integers; fixed set = {0,2,6,8}; common translate = n = 11; all-prime test = 11, 13, 17, 19 prime. The selected Prime quadruplet is this narrower member of the fixed-offset family.

Structural Tensions¶

T1: More specified prime positions vs local feasibility for unbounded recurrence. When an analyst adds an offset to H, the target asks for an additional prime at every occurrence. The added offset may also fill the final missing residue class modulo some p. Keeping it retains the richer target but obstructs arbitrarily large translates except possible small exceptions; changing or dropping it changes the target. Diagnostic: after adding an offset, does H still omit a residue modulo every prime relevant to the local test, and is the goal a single finite occurrence or a candidate for unbounded recurrence? This pressure belongs to pattern selection, not a proof that an admissible target recurs infinitely often.^[1]^[2]

Structural–Framed Character¶

This object lies near the structural end of the structural–framed spectrum. Its integer carrier, fixed set, one common shift, and all-prime predicate supply repeatable recognition criteria. It is not defined by a particular observer, instrument, nation, or organization. A mathematical community chooses notation and studies particular H, but neither authorship nor a formal institutional designation is required for an occurrence. Human practice matters to which tuples are named or investigated; it does not alter whether a specified n+H is all prime. The object has no built-in evaluative success criterion such as density or usefulness: recurrence and predicted abundance are separate questions. Importing the recognition test into the twin and quadruplet cases preserves all four roles, while importing one case’s exact gaps, residue images or conjectured frequency into the other would fail. Its character: a formally recognized arithmetic configuration with source-bound, nonconstitutive recurrence diagnostics.^[1]^[2]

Structural Core vs. Domain Accent¶

The most general skeleton is a finite set H, a common translation n+H, and a predicate applied to every translated member. The Set and Membership constituent accounts for H as an extensional collection: identifiable offsets, bivalent membership, a collection used in reasoning, distinct elements, and no dependence on listing order or repetition. It is the sole strict child→parent edge. A generic set, however, need not have integer shifts or an all-prime occurrence predicate.^[1]^[2]

The domain accent is the prime-number predicate and congruence obstruction. A future Prime would need independent substrates that genuinely fill the full translate-plus-all-predicate signature, with a useful shared mechanism and boundaries; resemblance to an ordinary “pattern” is insufficient. The live Pattern signature includes recurrence, evidence and transfer commitments not entailed by one finite tuple. Thus the arithmetic configuration remains domain-specific under the current reviewed identity.

This entry is part of Set and Membership.

In every case, a prime k-tuple contains a Set and Membership structure as a part: the finite offset set H. For H = {0,2} and H = {0,2,6,8}, membership is determinate, the collection is reasoned about as a whole, element identity persists, and permutation or repeated listing does not change the set. Set and Membership does not itself require a common prime translate, so a prime k-tuple is narrower.

Pattern is a related word, but Pattern as the encyclopedia fully defines it is not broader. Prime Triplet and Sexy Primes are narrower neighbors, not broader than every k-tuple: k can be two or four, and a six-unit pair gap is not universal. These distinctions are about how the entries are defined, not a claim that the narrower arithmetic cases fail to be fixed-offset tuples.

Relationships to Other Abstractions¶

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

Prime k-tuple → Set and Membership

Neighborhood in Abstraction Space¶

Prime k-tuple sits in a sparse region of the domain-specific corpus (69^th 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¶

Do not equate admissible H with an observed prime occurrence; n = 12 fails for admissible {0,2}. Do not equate nonadmissible H with “no occurrence”; n = 3 realizes {3,5,7} for {0,2,4}. Do not read Hardy and Littlewood’s conditional Theorem X1 as an unconditional asymptotic, or Maynard’s Theorem 1.2 as proof for every individually chosen H. Neither original supports a present-day global claim about exactly which fixed patterns have proven infinite recurrence.^[1]^[2]

References¶

[1] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[2] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p