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: v4 · 2026-09-19 · History
Domain-specific #: 1647
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Multiplicative Number Theory → Mathematics
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.

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The Pretend-Perfect Number

Some special numbers are called perfect because the smaller numbers that divide them add up to exactly the number, like 6 = 1 + 2 + 3. A Descartes number is a pretend-perfect number. It only works out if you cheat by pretending that one of its building blocks can't be split up, when really it can.

Fake Odd Perfect Number

A perfect number is one where all the numbers that divide it evenly, except itself, add up to exactly the number; 6 works because 1 + 2 + 3 = 6. For big numbers, mathematicians compute this by breaking the number into prime building blocks. A Descartes number, also called a spoof odd perfect number, is an odd number that seems to pass this test only because one of its building blocks is treated as prime when it is not. Descartes found an example in 1638 where the piece 22021 is treated as prime, even though 22021 = 19 x 19 x 61. Since the pretend prime is not really prime, the number is not actually perfect.

Spoof Odd Perfect Number

Let sigma(n) be the sum of all divisors of n. A number is perfect if sigma(n) = 2n. A Descartes number is an odd number D = km, with k and m greater than 1 and sharing no common factor, such that sigma(k)(m + 1) = 2D. Here m is the 'spoof prime': the formula uses m + 1, which would be m's divisor sum if m were prime, even though m is actually composite. Descartes' example was D = 3^2 x 7^2 x 11^2 x 13^2 x 22021, where 22021 = 19^2 x 61 is not prime, so D only fakes perfection. Because the divisor-sum function is multiplicative, spoofs show what the factor equations alone allow when primality is dropped.

A Descartes number, or spoof odd perfect number, is an odd integer D = km with k, m > 1 and gcd(k, m) = 1 such that sigma(k)(m + 1) = 2km = 2D, where the composite factor m is treated as a 'spoof prime' by using m + 1, the divisor sum a prime m would have, in place of the true sigma(m). By multiplicativity of sigma, D would then satisfy sigma(D) = 2D if m were prime; because m is composite, D is not actually perfect. Descartes' 1638 example is D = 3^2 * 7^2 * 11^2 * 13^2 * 22021, with 22021 = 19^2 * 61. Dittmer generalized the idea to spoof perfect factorizations, extending the geometric-series factor associated with a prime power to 'quasi-prime' bases that need not be prime. The concept is useful because it separates what follows purely from the multiplicative divisor-sum equations from what genuinely depends on primality and unique prime factorization, which is relevant to the study of odd perfect numbers.

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) — outgoing typed relations

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 includes Factorization Prime

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

Hierarchy paths (9) — routes to 7 parentless roots

Descartes Number → is a kind of Natural Number → is a kind of Discreteness → is a kind of Boundary

Show alternative paths (8)

Neighborhood in Abstraction Space¶

Descartes Number sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Integer Properties & Number-Theoretic Sets (15 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25