Perfect number

Classify a positive integer as perfect when the sum of all its proper positive divisors equals the integer itself, equivalently when its divisor-sum is exactly twice the integer.

Version

v2 · 2026-08-30 · History

Domain-specific #

2467

Origin domain

number theory

Subdomain

divisor sums and multiplicative arithmetic

Core Idea

A perfect number is a positive integer \(n\) satisfying \(s(n)=n\), where \(s(n)\) sums the positive proper divisors of \(n\); equivalently, \(\sigma(n)=2n\). Prime factorization makes the divisor-sum function multiplicative, converting the balance condition into arithmetic restrictions; in the even case this yields exactly the Euclid-Euler form built from a Mersenne prime, while the odd case remains constrained but unresolved.

Its autonomous residual is the exact self-balance of one positive integer under the proper-divisor sum, not numerical aesthetic perfection, near equality, a multiperfect ratio other than two, an amicable cycle, or possession of many divisors.

Scope of Application

Perfect number applies when the analyst can specify a positive integer \(n\), its positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\) and establish that the complete set of positive divisors is accounted for under one convention and its sum satisfies the exact equality \(\sigma(n)=2n\). The entry gives mathematical definitions and theorem status, not a search recipe. Counts of known examples, largest records, and computational lower bounds are time-indexed evidence rather than identity-bearing facts.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because perfect is evaluative in ordinary language, while divisor sums may include or exclude the integer and related literatures use multiply perfect, semiperfect, and unitary-perfect variants. The disciplined statement is that the object counts as Perfect number exactly when the complete set of positive divisors is accounted for under one convention and its sum satisfies the exact equality \(\sigma(n)=2n\)

Manages Complexity

The abstraction compresses even perfect numbers, hypothetical odd perfect numbers, base representations, aliquot fixed points, semiperfect and multiperfect generalizations, Mersenne-prime constructions, and computational searches into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares divisor convention, parity, prime factorization, sigma value, aliquot sum, Mersenne exponent, primality certificate, odd-case necessary conditions, computational bound, and publication date and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a positive integer \(n\), its positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\) and reject examples from a different problem. 2. Lock the rule. Express that the complete set of positive divisors is accounted for under one convention and its sum satisfies the exact equality \(\sigma(n)=2n\) independently of one notation or implementation. 3.

Knowledge Transfer

Transfer within number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from The positive proper divisors of 6 are 1, 2, and 3, whose sum is 6; equivalently, all positive divisors sum to 12. to The Euclid-Euler theorem states that every even perfect number, and only an even perfect number, has form \(2^{p-1}(2^p-1)\) with \(2^p-1\) prime. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Perfect number Domain-specific

Parents (1) — more general patterns this builds on

• Perfect number is a kind of Constraint Prime

  The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

• Perfect number → Constraint

Neighborhood in Abstraction Space

Perfect number sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

• Quasiperfect number — 0.95
• Deficient number — 0.93
• Sublime number — 0.92
• Hyperperfect number — 0.91
• Euler's totient function — 0.90

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