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: v2 · 2026-09-19 · History
Domain-specific #: 4435
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

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.

How would you explain it like I'm…

The Always-New Prime Chain

Start with the number 2. Multiply together all the numbers you have so far, add one, and break the answer into its smallest unbreakable piece (a prime); that piece is your next number. Then do it again and again. Each new number is always one you haven't had before. That list is the Euclid–Mullin sequence.

Multiply, Add One, Pick Smallest

The Euclid–Mullin sequence starts at 2. At each step, multiply every number in the list so far, add 1, and find the smallest prime that divides the result; that prime is the next term. So: 2, then 2 + 1 = 3, then 2 × 3 + 1 = 7, then 2 × 3 × 7 + 1 = 43, then 2 × 3 × 7 × 43 + 1 = 1807 = 13 × 139, so the next term is 13. None of the earlier primes can divide the 'plus one' number, because they all leave remainder 1. So every new term is a prime you haven't seen yet.

Least-Factor Euclid Sequence

The Euclid–Mullin sequence begins with 2 and is built by a fixed rule: take the product of all terms so far, add 1, and choose its least prime factor as the next term. Euclid's argument guarantees this factor is new, because each earlier prime divides the product and so leaves remainder 1 on the plus-one number. The first terms are 2, 3, 7, 43, 13, so the sequence isn't in increasing order. If you allowed any prime factor instead of the smallest, you'd get many branching possible sequences; picking the smallest makes one definite path. Changing the starting term or the selection rule gives a different sequence.

The Euclid–Mullin sequence is defined by a_1 = 2 and a_{n+1} = the least prime factor of a_1 a_2 ... a_n + 1. By Euclid's argument, each a_i divides the cumulative product and therefore cannot divide the product plus one, so every new term is a prime distinct from all earlier terms. This multiplicative memory excludes all previous primes from the next candidate. Without a selection rule, choosing any prime factor at each step generates a branching tree of Euclid sequences; least-prime-factor selection picks one deterministic branch. The identity of the sequence is fixed by three things together: the initial term 2, the cumulative-product-plus-one operation, and the least-prime-factor choice. Altering any of these (another start, the largest factor, or a different shift) yields a neighboring but different object.

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.
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) — outgoing typed relations

Euclid–Mullin sequence is a kind of Recursion Prime

What makes it its own entry: the domain-specific identity determined by the initial term, cumulative-product-plus-one operation, and least-prime-factor choice are fixed and every generated prime is distinct from earlier terms.

Hierarchy path (1) — routes to 1 parentless root

Euclid–Mullin sequence → is a kind of Recursion

Neighborhood in Abstraction Space¶

Euclid–Mullin sequence sits in a crowded region of the domain-specific corpus (12^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer & Divisor Number Theory (45 abstractions)

Nearest neighbors

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