Prime signature

Classify a positive integer by the unordered multiset of positive exponents in its unique prime factorization, discarding prime labels while preserving multiplicative shape.

Version

v2 · 2026-08-30 · History

Domain-specific #

2527

Origin domain

number theory

Subdomain

multiplicative classification of integers

Core Idea

If \(n=p_1^{a_1}\cdots p_k^{a_k}\) is the unique prime factorization of a positive integer with every \(a_i>0\), the prime signature of \(n\) is the unordered multiset \(\{a_1,\ldots,a_k\}\); the signature of one is the empty multiset. Unique factorization provides labeled prime-power coordinates, forgetting the prime bases and retaining only exponent multiplicities quotients integers by relabeling of their distinct prime factors, and symmetric functions of the exponents become invariants of each resulting class.

Its autonomous residual is the quotient invariant obtained by forgetting prime identities while retaining the complete multiplicity pattern of their exponents, not the prime factorization itself, a digital signature, the set of prime divisors, or a list that discards repeated exponents.

Scope of Application

Prime signature applies when the analyst can specify a positive integer with its unique prime factorization and the finite multiset of positive exponents attached to the distinct prime bases that divide it and establish that the object contains all and only the positive exponents from one integer's unique prime factorization, ignores which distinct primes carry them, preserves repetitions, and treats order as immaterial. The entry concerns positive integers and unordered nonzero prime exponents. Extensions to rationals, ideals, monoids, or ordered infinite exponent vectors require separately declared carriers and conventions.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because prime signature is not universally standardized; exponent pattern, prime bag, ordered signature, and full valuation vector can denote related but nonidentical objects. The disciplined statement is that the object counts as Prime signature exactly when the object contains all and only the positive exponents from one integer's unique prime factorization, ignores which distinct primes carry them, preserves repetitions, and treats order as immaterial

Manages Complexity

The abstraction compresses unordered and ordered signatures, sorted tuples and multisets, prime bags, partitions of total multiplicity, canonical least representatives, graph-isomorphism classes, and signature-based metrics 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 integer domain, factor count, exponent multiset, maximum exponent, minimum exponent, exponent sum, exponent gcd, divisor count, canonical ordering, and representative choice and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a positive integer with its unique prime factorization and the finite multiset of positive exponents attached to the distinct prime bases that divide it and reject examples from a different problem. 2. Lock the rule. Express that the object contains all and only the positive exponents from one integer's unique prime factorization, ignores which distinct primes carry them, preserves repetitions, and treats order as immaterial independently of one notation or implementation.

Knowledge Transfer

Transfer within number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from The integers \(12=2^2\cdot3\) and \(18=2\cdot3^2\) share signature \(\{2,1\}\), although their ordered exponent vectors relative to ascending primes differ. to Divisor Hasse diagrams of two integers with the same prime signature are isomorphic because each divisor corresponds to choosing one exponent between zero and the relevant factor exponent. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Prime signature Domain-specific

Parents (1) — more general patterns this builds on

• Prime signature is a kind of Classification Prime

  The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

• Prime signature → Classification

Neighborhood in Abstraction Space

Prime signature sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

• Quasiperfect number — 0.90
• Additive function — 0.90
• Integer factorization — 0.89
• Euler's totient function — 0.89
• Dedekind psi function — 0.89

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