Smooth Number

A positive integer is B-smooth when every prime divisor is at most the declared bound B, making its factorization lie entirely within a small-prime factor base.

Version

v2 · 2026-09-06 · History

Domain-specific #

2797

Origin domain

mathematics

Subdomain

number theory

Aliases

Friable number, B-smooth integer, Y-smooth number

Core Idea

A positive integer \(n\) is B-smooth if every prime divisor of \(n\) is at most \(B\); equivalently, its largest prime factor \(P^+(n)\le B\), with \(1\) included by convention. Smoothness is relative to the stated bound. The counting function \(\Psi(x,B)\) records B-smooth integers not exceeding \(x\).

The recognition invariant is positive integer + complete prime factorization constraint + explicit upper bound on every prime factor.

Scope of Application

Smooth numbers drive integer-factorization and discrete-logarithm algorithms because relations over a fixed factor base can be combined algebraically. They also appear in primality proving, cryptographic parameter assessment, fast Fourier transform sizes, scheduling/combinatorial enumeration, and analytic number theory.

Algorithm design balances a larger factor base, which increases the probability that sampled values are smooth, against more costly relation collection and linear algebra.

Clarity

Always attach the bound. “Smooth” without B is incomplete. B need not be prime; the condition uses the largest prime not exceeding it. The factor B itself need not divide n.

Power-smoothness, semismoothness (allowing one controlled larger factor), and smoothness relative to ideals or polynomials are related but distinct definitions.

Manages Complexity

Smoothness compresses a full factorization into a one-threshold predicate while retaining an exponent vector over a small factor base. Algorithms exploit that vector representation to turn multiplicative relations into linear algebra.

Abstract Reasoning

1. Declare n and B.
2. Factor n or certify bounds on its prime divisors.
3. Find the largest prime factor.
4. Compare it with B.
5. Record the factor-base exponent vector where relevant.
6. For sampled integers up to x, estimate \(\Psi(x,B)\) using an appropriate regime.
7. Balance smoothness probability against factor-base and linear-algebra costs.
8. Separate smooth, semismooth, and power-smooth variants.

Knowledge Transfer

The portable structure is membership determined by whether every atomic factor lies within an allowed vocabulary. The proposed immediate parent is Natural Number.

Relationships to Other Abstractions

Current abstraction Smooth Number Domain-specific

Parents (1) — more general patterns this builds on

• Smooth Number is a kind of Natural Number Domain-specific

  Natural Number is the proposed immediate parent.

Hierarchy paths (8) — routes to 6 parentless roots

• Smooth Number → Natural Number → Discreteness → Boundary

Show alternative paths (7)

Neighborhood in Abstraction Space

Smooth Number sits in a sparse region of the domain-specific corpus (82^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Additive Number Theory & Series Tests (6 abstractions)

Nearest neighbors

• Giuga Number — 0.83
• Fundamental Theorem of Arithmetic — 0.82
• Refactorable number — 0.82
• Descartes Number — 0.81
• Schnirelmann Density — 0.81

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