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.
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. Inclusion test: Require two proven primes with absolute difference exactly six; for tuples, verify every listed arithmetic-progression term. Exclusion test: Exclude cousin primes gap four, twin primes gap two, composite endpoint pairs, and numbers merely congruent modulo six. Nearest boundary: Sexy primes name a fixed prime gap; admissible prime tuples study broader offset patterns and divisibility obstructions. Exit condition: Membership fails immediately when either endpoint is composite or the gap is not six. Common misclassifications: 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. Nearest named distinctions: Twin primes: Differ by two. Cousin primes: Differ by four. Prime sextuplet: Names a different multi-offset constellation. Numbers congruent mod six: Need not be prime or differ by exactly six.
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.
Relationships to Other Abstractions¶
Current abstraction Sexy Primes Domain-specific
Parents (1) — more general patterns this builds on
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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.
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