Euclid–Mullin sequence¶

A recursively defined prime sequence taking each new term as the least prime factor of one plus the product of all preceding terms.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4435
Origin domain: number theory
Subdomain: number theory

Core Idea¶

Starting from two, form one plus the cumulative product and choose its least prime divisor at every step; Euclid’s argument ensures this divisor is new. Multiplicative memory excludes every prior prime from the next candidate integer, while least-factor selection makes one deterministic branch through an otherwise branching Euclid sequence tree. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Euclid–Mullin sequence belongs to number theory and is useful where the analyst can specify the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms. The scope is broad within that domain but bounded by the need for the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Euclid–Mullin sequence can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Euclid–Mullin sequence. Euclid–Mullin sequence compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Multiplicative memory excludes every prior prime from the next candidate integer, while least-factor selection makes one deterministic branch through an otherwise branching Euclid sequence tree., and type the carrier, state every parameter and convention in the definition, test that the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Euclid–Mullin sequence Domain-specific

Parents (1) — more general patterns this builds on

Euclid–Mullin sequence is a kind of Recursion Prime

The proposed strict upward parent is prime:recursion.

Hierarchy path (1) — routes to 1 parentless root

Euclid–Mullin sequence → Recursion

Neighborhood in Abstraction Space¶

Euclid–Mullin sequence sits in a crowded region of the domain-specific corpus (21^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

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