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.^[1]

Descartes' 1638 example is.

D = 3² · 7² · 11² · 13² · 22021,

where 22021 = 19² · 61 is composite. If 22021 were prime, multiplicativity of the divisor-sum function would make σ(D)=2D; because it is not prime, the actual integer is not perfect. Dittmer formalizes the broader object as a spoof perfect factorization by extending the geometric-series factor associated with a prime power to “quasi-prime” bases that need not be prime.^[2]

The abstraction is useful because it isolates how much of odd-perfect-number structure follows from the multiplicative factor equations alone and how much genuinely depends on primality and unique prime factorization.

Structural Signature¶

the odd candidate integer — an integer D>1 with no factor 2;
the coprime split — D=km, with k,m>1 and gcd(k,m)=1;
the genuine factor block — k is evaluated with the ordinary divisor-sum function σ(k);
the spoof-prime block — composite m is assigned the prime-like contribution m+1 rather than its actual divisor sum;
the spoof perfection equation — σ(k)(m+1)=2km;
the failed literal perfection test — because m is composite, the actual σ(D) need not equal 2D;
the factorization-sensitive identity — the witness is not merely the integer value; it includes which factor is designated as spoof prime.

Some literature generalizes from positive integer Descartes numbers to spoof factorizations with composite or even negative quasi-prime bases. This entry retains the positive odd-integer identity and treats those constructions as extensions.

What It Is Not¶

Not an odd perfect number. No odd perfect number is known. A Descartes number satisfies a relaxed factor equation, not σ(D)=2D under the actual divisors.
Not a pseudoperfect/semiperfect number. Those terms concern expressing an integer as a sum of some proper divisors, a different property.
Not an almost perfect number. Almost perfect numbers satisfy σ(n)=2n-1; the resemblance in naming does not identify the class.
Not any near miss to perfection. The defining error is structured: one composite factor is evaluated using the prime formula m+1.
Not just Descartes' single integer. The term names a formally defined class/witness structure, even though the historical positive example is exceptionally constrained.
Not a prime number. The “spoof prime” designation is part of the relaxed computation and explicitly does not assert primality.

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.^[2] 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.^[3] These are statements about the generalized research object, not evidence that additional positive Descartes integers satisfy every narrower convention.

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.

Therefore the phrase “would be perfect if 22021 were prime” is a counterfactual shorthand for an exact arithmetic equation. It does not mean a primality test was uncertain, nor that changing the number while retaining its value could change its divisors.

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.

It also provides a search representation. Algorithms enumerate candidate bases, exponents, and spoof-prime positions satisfying a rational product equation, then apply primitiveness, sign, coprimality, and genuine-primality checks. A rare historical curiosity becomes a tractable family of Diophantine witnesses.

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.

Factorization audit. Record the designated spoof factor. Recombining or splitting bases may alter whether a factorization is primitive or even qualifies under a chosen generalized definition.

Constraint transfer. Test which lemmas about odd perfect numbers use only the factor equation and which invoke primality. Only the former can transfer automatically to spoofs.

Search bounding. Use congruences, size inequalities, exponent parity, and minimum-factor results to prune quasi-prime tuples before exact arithmetic verification.

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.

Examples¶

Descartes' example. 3²·7²·11²·13²·22021 satisfies the spoof equation when 22021 contributes as a prime, although 22021=19²·61 and the number is not actually perfect.^[2]

Negative-base generalization. Voight's factorization with a negative quasi-prime base satisfies a generalized spoof equation. It demonstrates extension of the factorization formalism but is not a positive Descartes number under this entry's locked scope.^[3]

Rejected candidate. An odd integer with σ(D) numerically close to 2D but no coprime (k,m) witness satisfying the spoof equation is not a Descartes number.

Structural Tensions¶

T1: Integer versus witnessed factorization. The property is often named as a number class, but generalized work treats the factorization as primary. Diagnostic: state the designated spoof base and convention.

T2: Relaxation versus accidental overclaim. The spoof equation preserves a formal identity while literal perfection fails. Diagnostic: calculate actual σ(D) separately.

T3: Historical narrowness versus modern generalization. Positive Descartes numbers, negative-base spoofs, and multiperfect spoofs are related but not interchangeable. Diagnostic: declare allowed signs, exponents, and target multiplier.

T4: Base list versus prime factorization. Quasi-prime bases may be composite, so standard uniqueness arguments do not apply directly. Diagnostic: identify which steps require genuine primes.

T5: Structural analogy versus solution evidence. Spoofs illuminate constraints around odd perfect numbers but are not candidate odd perfect numbers. Diagnostic: locate the exact place primality is missing.

Structural–Framed Character¶

Descartes Number is almost wholly structural: qualification is determined by exact arithmetic. The small framed component is terminological—authors vary between number-centered and factorization-centered definitions and in which generalizations they admit.

Structural Core vs. Domain Accent¶

The portable core is factorization plus a deliberately relaxed atomicity assumption. The domain accent is the ordinary divisor-sum function, oddness, coprime split, prime-like m+1 contribution, and perfect-number equation. Removing these yields general assumption relaxation or factorization; retaining them keeps the node inside number theory. It is domain-specific, not prime.

factorization: the witness decomposes D into blocks whose formal contributions multiply.
natural_number: positive Descartes numbers are a specialized class of natural numbers carrying an additional factorization witness.
canonical_form: prime-power form motivates the spoof representation, but allowing composite bases weakens canonical uniqueness.

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

Not to Be Confused With¶

odd perfect number;
even perfect number;
pseudoperfect or semiperfect number;
almost perfect number;
Erdős–Nicolas number;
arbitrary spoof perfect factorization with negative bases;
spoof multiperfect number with multiplier other than two.

References¶

[1] Dris, Jose Arnaldo B., and Florian Luca. “The Non-Euler Part of a Spoof Odd Perfect Number Is Not Almost Perfect.” 2015. https://arxiv.org/abs/1503.03860 registry ↩

[2] Dittmer, Samuel J. “Spoof Odd Perfect Numbers.” Mathematics of Computation 83, no. 289 (2014): 2575–2582. https://doi.org/10.1090/S0025-5718-2013-02793-7 registry ↩a ↩b ↩c

[3] BYU Computational Number Theory Group. “Odd, Spoof Perfect Factorizations.” Journal of Number Theory 234 (2022): 31–47. https://doi.org/10.1016/j.jnt.2021.07.028 registry ↩a ↩b