Regular prime¶
An odd prime p that does not divide the class number of the p-th cyclotomic field, equivalently one that divides none of the relevant Bernoulli-number numerators.
Core Idea¶
A regular prime is an odd prime \(p\) that does not divide the class number of the cyclotomic field \(\mathbb{Q}(\zeta_p)\). Equivalently, by Kummer’s criterion, p divides none of the numerators of \(B_2,B_4,\ldots,B_{p-3}\). The prime 2 is often included as regular by convention. Failure at even index \(2k\) makes \((p,2k)\) an irregular pair; the number of such failures is the index of irregularity.
Scope of Application¶
The classification links cyclotomic ideal theory, Bernoulli-number arithmetic, and Kummer’s proof of Fermat’s Last Theorem for regular prime exponents. Irregular pairs record the indices at which the criterion fails. Conjectured density near \(e^{-1/2}\) is not part of the definition or a proof of infinitude.
- Cyclotomic fields. Class-number divisibility supplies the original algebraic-number-theory criterion.
- Bernoulli arithmetic. Kummer’s equivalent test turns regularity into finitely many numerator divisibility checks.
- Fermat’s Last Theorem. Regular prime exponents satisfy Kummer’s theorem.
- Computational number theory. Tables record irregular pairs and indices, not merely a binary label.
Clarity¶
State whether p is odd, which Bernoulli-number convention is used, and whether 2 is included by convention. For an individual p, provide either the class-number nondivisibility or all relevant Bernoulli checks; one positive divisibility produces an irregular pair and refutes regularity. The closest near miss sets the boundary: An irregular prime is the exact complement among odd primes: at least one relevant divisibility obstruction occurs.
Manages Complexity¶
The term compresses a deep equivalence between an ideal-class invariant of a cyclotomic field and a finite pattern of Bernoulli-number divisibility. Irregular pairs retain the locations of failure that a binary label would discard. The central class-number criterion–Bernoulli criterion tradeoff is this: Equivalent definitions expose different mathematical structures and computational costs. A second binary regularity–irregularity index tension matters because The label records absence or presence, while pairs retain how many and where failures occur.
Abstract Reasoning¶
Use three linked moves: verify that p is prime and handle p=2 under the stated convention; construct the p-th cyclotomic setting or enumerate even Bernoulli indices 2 through p−3; test divisibility of the class number or every relevant Bernoulli numerator by p. As a collapse test, the case exits as soon as p divides the class number or one relevant Bernoulli numerator. A fourth check is to declare regular only if no obstruction appears; otherwise record each irregular pair.
Knowledge Transfer¶
The equivalence transfers between algebraic and computational approaches inside number theory. Similar labels such as Euler-regular replace Bernoulli numbers with another sequence and therefore define related but distinct classes; everyday ‘regularity’ language does not transfer literally. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. The criteria partition odd primes into regular and irregular classes.
Neighborhood in Abstraction Space¶
Regular prime sits in a sparse region of the domain-specific corpus (63rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Square-Free Integer — 0.87
- Quartic reciprocity — 0.85
- Covering Set — 0.85
- Fermat's Little Theorem — 0.84
- Jordan's totient function — 0.84
Computed from structural-signature embeddings · 2026-10-08