Sexy Primes¶
A pair of prime numbers separated by exactly six, with triplet and higher variants requiring repeated six-unit prime gaps.
Core Idea¶
Sexy primes are fixed-gap prime pairs: p and p+6 must both be prime. The terminology is a pun on the Latin word for six, not a mathematical property beyond the gap and endpoint primality.
Longer sexy-prime constellations add p+12, p+18, and so on, with every term separately prime. Small-prime congruences constrain which tuple patterns can occur.
Structural Signature¶
Sig role-phrases:
- Lower integer p — Provides first candidate. It is first endpoint. Counterfactual: Composite p rejects the pair.
- Offset six — Defines the exact arithmetic separation. It is relation. Counterfactual: Any other prime gap is another class.
- Upper integer p+6 — Provides second candidate. It is second endpoint. Counterfactual: Its primality is independent.
- Primality tests — Verify both endpoints over positive integers. It is membership test. Counterfactual: Probable-prime status may need qualification.
- Ordered/unordered convention — Prevents duplicate listing of the same pair. It is representation. Counterfactual: Reversing endpoints does not create a new pair.
- Tuple extension — Requires each successive p+6k to be prime. It is subtype. Counterfactual: A pair alone does not establish a triplet.
What It Is Not¶
- They are not twin primes.
- A gap of six alone does not establish primality.
- One pair does not imply a triplet.
- Probable primes should not be reported as proven without qualification.
- Closest near-miss. Sexy primes name a fixed prime gap; admissible prime tuples study broader offset patterns and divisibility obstructions.
Scope of Application¶
- Number theory. Studies fixed prime gaps.
- Sieve theory. Counts admissible constellations.
- Computational mathematics. Searches and certifies examples.
- Integer sequences. Catalogs pairs and tuples.
Clarity¶
State endpoint convention, exact gap, primality proof/probable status, tuple length, offsets, search bound, arithmetic precision, duplicate handling, and software/certificate for large claims.
Manages Complexity¶
A tiny relational definition opens into deep distribution, admissibility, and certification questions about prime constellations.
Abstract Reasoning¶
- Choose lower candidate p.
- Test p and p+6 for primality.
- Record pair once in increasing order.
- For tuples test every added six-offset.
- Qualify computational evidence and bounds.
Knowledge Transfer¶
The definition transfers exactly across integer ranges, but computational claims require compatible primality standards, bounds, and exhaustive-search guarantees.
Examples¶
Canonical¶
5 and 11 are both prime and 11−5=6, so they form a sexy-prime pair.
Mapped back: p → 5 prime; offset → 6; q → 11 prime; verdict → pair.
Applied / In Practice¶
7 and 13 differ by six and are prime, but 7,13,19 is a triplet only after 19 is separately verified prime.
Mapped back: pair → yes; third → requires test; tuple → not inferred.
Structural Tensions¶
T1 — Simple Gap Rule versus Hard Primality. The relation is elementary while finding large examples depends on expensive prime certification.
Diagnostic: Are endpoints proven or merely probable primes?
T2 — Pair Abundance versus Tuple Obstruction. Repeated six-gaps encounter modular divisibility patterns that constrain longer runs.
Diagnostic: Which offset constellation is admissible modulo small primes?
Structural–Framed Character¶
Sexy Primes are structural as a fixed-gap predicate on prime endpoints.
Structural Core vs. Domain Accent¶
The core is two integers, primality, and difference six; number theory supplies gaps, tuples, sieves, and certification.
Instantiates / Related Primes¶
This entry presupposes Classification.
-
Approved root. No reviewed parent entails this prime pair class.
-
Related — twin primes, cousin primes, prime gap, admissible tuple, and primality test. They provide neighboring gaps, broader relation, extension, and verification.
Relationships to Other Abstractions¶
Current abstraction Sexy Primes Domain-specific
Parents (1) — more general patterns this builds on
-
Sexy Primes presupposes Classification Prime
Sexy Primes presupposes Classification because the class is established by the exact six-unit gap rule over pairs of primes.Every reviewed Sexy Primes instance depends on the parent role: the class is established by the exact six-unit gap rule over pairs of primes. Removing that role makes the frozen child identity undefined or changes it into a different abstraction. Classification can occur without Sexy Primes, so the relation is dependency rather than subsumption.
Hierarchy path (1) — routes to 1 parentless root
- Sexy Primes → Classification
Neighborhood in Abstraction Space¶
Sexy Primes sits in a crowded region of the domain-specific corpus (35th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Number & Formal Language Properties (7 abstractions)
Nearest neighbors
- Achilles Number — 0.91
- Wagstaff Prime — 0.90
- Power Residue Symbol — 0.88
- Maximum subarray problem — 0.88
- Dyadic Rational — 0.87
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Twin primes. Tell: Differ by two.
- Cousin primes. Tell: Differ by four.
- Prime sextuplet. Tell: Names a different multi-offset constellation.
- Numbers congruent mod six. Tell: Need not be prime or differ by exactly six.
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Sexy_primes (revision 1359436125).
- Preserved source candidate: http://www.numberphile.com/videos/sexy_primes.html
- Preserved source candidate: https://web.archive.org/web/20181023181320/http://numberphile.com/videos/sexy_primes.html
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.