Wilson quotient¶

For a prime p, the integer ((p−1)!+1)/p, whose residues encode refinements of Wilson's theorem and define Wilson primes when divisible by p.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7493
Origin domain: number theory
Subdomain: specialized structures

Core Idea¶

The Wilson quotient normalizes the divisibility guaranteed by Wilson's theorem into an arithmetic invariant of a prime. Factorial congruence makes the numerator divisible by p; examining the resulting quotient modulo p detects the exceptional Wilson-prime condition. 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.

The load-bearing residual is not the broad topic of number theory. It is For a prime p, the integer ((p−1)!+1)/p, whose residues encode refinements of Wilson's theorem and define Wilson primes when divisible by p.

Scope of Application¶

Wilson quotient belongs to number theory and is useful where the analyst can specify a positive integer p, factorial, divisibility by p, quotient, modular residue, Bernoulli congruences and Wilson-prime condition, then evaluate p is prime under the standard integer definition and the quotient and any congruence use one declared modulus. The scope is broad within that domain but bounded by the need for p is prime under the standard integer definition and the quotient and any congruence use one declared modulus. 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 p is prime under the standard integer definition and the quotient and any congruence use one declared modulus 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 Wilson quotient 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 Wilson quotient. Wilson quotient 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: a positive integer p, factorial, divisibility by p, quotient, modular residue, Bernoulli congruences and Wilson-prime condition. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express p is prime under the standard integer definition and the quotient and any congruence use one declared modulus independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a positive integer p, factorial, divisibility by p, quotient, modular residue, Bernoulli congruences and Wilson-prime condition, Factorial congruence makes the numerator divisible by p; examining the resulting quotient modulo p detects the exceptional Wilson-prime condition., and type the carrier, state every parameter and convention in the definition, test that p is prime under the standard integer definition and the quotient and any congruence use one declared modulus, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Wilson quotient Domain-specific

Parents (1) — more general patterns this builds on

Wilson quotient is a kind of Recurrence Prime

The proposed strict upward parent is prime:recurrence.

Hierarchy path (1) — routes to 1 parentless root

Wilson quotient → Recurrence

Neighborhood in Abstraction Space¶

Wilson quotient sits in a crowded region of the domain-specific corpus (21^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

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