Descartes Number¶

An odd composite integer equipped with a coprime factorization D=km that satisfies the perfect-number divisor-sum equation only when the composite factor m is formally treated as a single prime-like factor.

Version: v3 · 2026-09-06 · History
Domain-specific #: 1647
Origin domain: mathematics
Subdomain: multiplicative number theory
Aliases: Spoof odd perfect number

Core Idea¶

A Descartes number—also called a spoof odd perfect number—is an odd integer that imitates the divisor-sum factorization of an odd perfect number because one composite factor is deliberately treated as if it were prime. In a common formulation, let D = km with k,m > 1, gcd(k,m)=1, and D odd. If.

σ(k)(m+1) = 2km = 2D,

then D is a spoof odd perfect number with m acting as the spoof prime. The equation uses m+1, the divisor sum that a genuine prime m would have, rather than the actual σ(m) when m is composite.

Scope of Application¶

Descartes numbers occur in research on perfect numbers, divisor sums, Diophantine equations, factorization constraints, and computational searches. They provide a controlled neighboring class to the unsolved odd-perfect-number problem. Researchers ask which Euler-like forms and divisibility restrictions survive the spoof relaxation, how many quasi-prime bases are required, and whether generalized negative-base or multiperfect spoofs exist.

Dittmer proved that Descartes' example is the only spoof odd perfect number with fewer than seven quasi-prime factors under his formalization. Later work broadened the factorization setting and found additional primitive spoof perfect factorizations, while proving finiteness for a fixed number of bases under stated conditions.

Clarity¶

The decisive distinction is between formal factor contribution and actual divisor sum. For a genuine prime power p^a, multiplicativity uses 1+p+...+p^a. A spoof factorization allows a base to occupy this formal slot even when it is not prime. In Descartes' example, the factor 22021 is assigned 1+22021; its internal factors 19² and 61 would contribute additional divisors in the actual σ(D) computation.

Manages Complexity¶

The spoof formulation separates two sources of constraint that ordinary prime factorization bundles together. The multiplicative perfection equation supplies algebraic relationships among bases and exponents. Primality supplies the right to interpret each geometric factor as an actual divisor sum. Relaxing the second while retaining the first lets researchers study the algebraic skeleton independently.

Abstract Reasoning¶

Witness verification. Given (k,m), check oddness, nontriviality, coprimality, compositeness of m, and the exact equation σ(k)(m+1)=2km.

Actual-versus-spoof comparison. Compute both σ(D) and the spoof expression. Equality of the spoof expression with 2D does not imply equality of σ(D) with 2D.

Knowledge Transfer¶

The complete identity transfers only within number theory and its computational implementations. Outside it, “pretend a composite component is atomic and see which global invariant survives” is a useful analogy, but it is not literally a Descartes number.

The structural residue includes factorization, model relaxation, and assumption auditing. Those patterns travel; the divisor-sum equation and spoof-prime witness do not.

Relationships to Other Abstractions¶

Current abstraction Descartes Number Domain-specific

Parents (2) — more general patterns this builds on

Descartes Number is a kind of Natural Number Domain-specific

natural_number: positive Descartes numbers are a specialized class of natural numbers carrying an additional factorization witness.
Descartes Number is part of Factorization Prime

factorization: the witness decomposes D into blocks whose formal contributions multiply.

Hierarchy paths (9) — routes to 7 parentless roots

Descartes Number → Natural Number → Discreteness → Boundary

Neighborhood in Abstraction Space¶

Descartes Number sits in a sparse region of the domain-specific corpus (84^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

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