Carmichael number¶

A composite integer that satisfies the Fermat congruence for every integer base, so it systematically imitates prime behavior under the basic Fermat test.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3597
Origin domain: computational number theory
Subdomain: computational number theory

Core Idea¶

A Carmichael number is a positive composite n such that b^n is congruent to b modulo n for every integer b; equivalently, b^(n-1) is congruent to one for every b coprime to n. Its square-free prime factors satisfy Korselt's divisibility criterion, making every coprime base pass the Fermat congruence even though the number is composite. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Carmichael number belongs to computational number theory and is useful where the analyst can specify the typed computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate n is composite, square-free, and every prime divisor p of n has p minus one dividing n minus one, equivalently the universal Fermat congruence holds. The scope is broad within that domain but bounded by the need for n is composite, square-free, and every prime divisor p of n has p minus one dividing n minus one, equivalently the universal Fermat congruence holds. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making n is composite, square-free, and every prime divisor p of n has p minus one dividing n minus one, equivalently the universal Fermat congruence holds the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Carmichael number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Carmichael number. Carmichael number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express n is composite, square-free, and every prime divisor p of n has p minus one dividing n minus one, equivalently the universal Fermat congruence holds independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of computational number theory because they reuse the typed computational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Its square-free prime factors satisfy Korselt's divisibility criterion, making every coprime base pass the Fermat congruence even though the number is composite., and type the carrier, state every parameter and convention in the definition, test that n is composite, square-free, and every prime divisor p of n has p minus one dividing n minus one, equivalently the universal Fermat congruence holds, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Carmichael number Domain-specific

Parents (1) — more general patterns this builds on

Carmichael number is a kind of Adversarial Signal Imitation Prime

The proposed strict upward parent is prime:adversarial_signal_imitation.

Hierarchy path (1) — routes to 1 parentless root

Carmichael number → Adversarial Signal Imitation → Signaling → Information Asymmetry → Asymmetry

Neighborhood in Abstraction Space¶

Carmichael number sits in a crowded region of the domain-specific corpus (18^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08