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.

Version

v1 · 2026-09-28 · History

Domain-specific #

11730

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Algebraic Number Theory, Cyclotomic Fields → Mathematics

Core Idea

A regular prime is an odd prime \(p\) that does not divide the class number of the cyclotomic field \(\mathbb{Q}(\zeta_p)\). Kummer proved an equivalent computational criterion: \(p\) divides none of the numerators of the Bernoulli numbers \(B_2,B_4,\ldots,B_{p-3}\). The prime 2 is often included as regular by convention rather than by the odd-prime test as stated.

Failure at even index \(2k\) makes \((p,2k)\) an irregular pair; the number of such failures is the index of irregularity. Thus regularity is an absence-of-obstruction classification, not a vague claim that a prime behaves typically.

Kummer introduced the distinction in work on Fermat’s Last Theorem and proved the theorem for regular prime exponents. The conjectured asymptotic density near \(e^{-1/2}\) is a heuristic/conjectural population claim, not part of the definition and not a proof that infinitely many regular primes exist.

Structural Signature

Sig role-phrases:

• Prime p. Supplies the odd prime being classified and determines the cyclotomic field and Bernoulli range. Constitutive carrier. If altered: A composite integer is outside the classification even if analogous divisibilities are tested.
• Cyclotomic class number. Measures ideal classes in the ring of integers of Q(ζp). One equivalent diagnostic. If altered: Divisibility by p changes the classification from regular to irregular.
• Relevant Bernoulli numerators. Provide the finite list B2 through Bp−3 at even indices. Equivalent computational diagnostic. If altered: One p-divisible numerator is sufficient for irregularity.
• Regularity verdict. Records absence of the specified p-divisibility and supports Kummer’s arithmetic consequences. Classification output. If altered: The verdict is criterion-bound and does not mean prime numbers are ‘regular’ in an everyday sense.

What It Is Not

• Not a generic prime number. Regular primes form a criterion-defined subclass.
• Not the complement of every ‘irregular prime’ usage. Bernoulli or B-irregularity must be distinguished from Euler-irregular and other generalizations.
• Not defined by Fermat’s Last Theorem. Kummer’s theorem motivates the class but the divisibility criteria define it.
• Not established by density heuristics. A conjectured frequency does not classify an individual prime or prove infinitude.

Scope of Application

The classification connects cyclotomic ideal theory, Bernoulli divisibility, and classical work on Fermat equations.

• 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.

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.

Abstract Reasoning

1. Verify that p is prime and handle p=2 under the stated convention.
2. Construct the p-th cyclotomic setting or enumerate even Bernoulli indices 2 through p−3.
3. Test divisibility of the class number or every relevant Bernoulli numerator by p.
4. Declare regular only if no obstruction appears; otherwise record each irregular pair.
5. Keep Kummer’s FLT consequence and density conjectures downstream from the classification.

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.

Examples

Canonical

For a candidate odd prime p, compute the relevant Bernoulli numerators; if none is divisible by p, Kummer’s criterion classifies p as regular.

Mapped back: prime p → the tested odd prime; cyclotomic class number → equivalently not p-divisible; relevant Bernoulli numerators → B2 through Bp−3; regularity verdict → regular when every test is negative.

Applied / In Practice

If p divides the numerator of B2k for an allowed index, (p,2k) is recorded and p is irregular regardless of the other indices.

Mapped back: prime p → the divisor; cyclotomic class number → p-divisible by equivalence; relevant Bernoulli numerators → the failing B2k; regularity verdict → irregular with a recorded pair.

Structural Tensions

T1: class-number criterion vs. Bernoulli criterion. Equivalent definitions expose different mathematical structures and computational costs. Diagnostic: Which criterion supplies the evidence, and are conventions aligned?

T2: binary regularity vs. irregularity index. The label records absence or presence, while pairs retain how many and where failures occur. Diagnostic: Is the binary verdict enough for the downstream theorem?

T3: empirical density vs. unproved infinitude. Large computations can match a heuristic density without proving the asymptotic claim. Diagnostic: Is the statement observed, conjectured, or proved?

Structural–Framed Character

Regular prime is strongly structural within number theory. Evaluative weight: none; ‘regular’ is a technical classification. Human-practice-bound: the name and treatment of 2 are conventional, while divisibility consequences are formal. Institutional origin: algebraic number theory stabilizes the criteria. Vocabulary travels: class numbers and Bernoulli divisibility travel within arithmetic. Import versus recognize: Euler-regular analogues deliberately change the sequence and must be relabeled. Its character: an equivalence-backed absence-of-obstruction class of primes.

Structural Core vs. Domain Accent

Skeletal core. A carrier belongs to a subclass exactly when none of a finite family of obstructions occurs, with two equivalent diagnostic representations.

Domain-bound accent. The carrier is an odd prime, the obstructions are divisibility of a cyclotomic class number or Bernoulli numerators, and Kummer’s theorem supplies the historical consequence.

Why not prime. Obstruction-free classification is portable, but the specific equivalence and arithmetic objects make regular prime a number-theoretic species rather than a cross-domain abstraction.

Instantiates / Related Primes

• Classification. The criteria partition odd primes into regular and irregular classes.
• Equivalence. Class-number and Bernoulli tests yield the same verdict.
• Obstruction. A single irregular pair witnesses failure of regularity.
• No new DAG edge is asserted during repair.

Neighborhood in Abstraction Space

Regular prime sits in a sparse region of the domain-specific corpus (63^rd 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

Not to Be Confused With

• Prime number. Tell: Regularity adds a cyclotomic/Bernoulli divisibility condition.
• Irregular prime. Tell: This is the exact odd-prime complement under Kummer’s criterion.
• Euler-regular prime. Tell: That generalization tests Euler-number divisibility rather than the Bernoulli criterion.
• Wolstenholme prime. Tell: It is a different rare divisibility class even though special irregular pairs can be related historically.

References

• Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Regular_prime (revision 1365822533).
• Preserved source candidate: https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/
• Preserved source candidate: http://tau.ac.il/~corry/publications/articles/pdf/Computers%20and%20FLT.pdf
• Preserved source candidate: https://www.ams.org/journals/proc/1954-005-02/S0002-9939-1954-0061124-6/S0002-9939-1954-0061124-6.pdf
• Preserved source candidate: http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002191873
• Preserved source candidate: https://archive.org/details/elementaryanalyt0000nark/page/475
• Preserved source candidate: https://primes.utm.edu/top20/page.php?id=25
• Preserved source candidate: http://www.digizeitschriften.de/resolveppn/GDZPPN002146738
• Preserved source candidate: https://projecteuclid.org/euclid.jmsj/1260541355

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.