Euclid number¶

An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4431
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

A Euclid number of the first kind has form E_n=p_n#+1; authors call p_n#−1 a Euclid number of the second kind or Kummer number. Adding or subtracting one makes the number coprime to every prime included in the primorial, though the resulting number need not itself be prime. 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…

Multiply Primes, Add One

Some numbers can't be split into equal groups except in boring ways; they're called primes, like 2, 3 and 5. If you multiply the first few primes together and add one, you get a Euclid number. None of those primes can share it out evenly, because there's always one left over. But the new number isn't always a prime itself.

Prime Product Plus One

A Euclid number is made by multiplying the first few prime numbers in order and then adding 1. For example, 2 × 3 × 5 = 30, and 30 + 1 = 31. Dividing a Euclid number by any of the primes you used always leaves a remainder of 1. But be careful: the new number does not have to be prime itself; it just can't be divided by those first primes. Some people also use subtracting 1 instead and call that a second kind.

Primorial Plus or Minus One

The primorial p_n# is the product of the first n primes, and the Euclid number of the first kind is E_n = p_n# + 1. Some authors also call p_n# − 1 a Euclid number of the second kind, or a Kummer number. Adding or subtracting one makes the result coprime to every prime in the product, since each leaves remainder ±1. That idea comes from Euclid's proof that there are infinitely many primes, but the proof never claims E_n is prime, and 30031 = 59 × 509 shows it can fail. Multiplying an arbitrary collection of primes and adding one gives a broader 'Euclid-style' number, not a Euclid number in this strict sense.

Let p_n# denote the primorial, the product of the first n consecutive primes. The Euclid number of the first kind is E_n = p_n# + 1, and p_n# − 1 is called by some authors a Euclid number of the second kind or a Kummer number. Because each prime p_i with i ≤ n divides p_n#, it cannot divide p_n# ± 1, so the result is coprime to all those primes. Euclid's infinitude-of-primes argument motivates the construction, but the argument does not claim E_n is prime; it only shows that any prime factor of E_n lies outside the first n primes. The definition is specific: the product must run over the consecutive initial primes, and the sign convention must be stated. Products over arbitrary finite prime sets plus or minus one are a broader family of Euclid-style numbers, not Euclid numbers in this sense.

Scope of Application¶

Euclid number belongs to number theory and is useful where the analyst can specify the ordered primes, nth primorial, plus-or-minus-one convention, integer factorization, and divisibility relations, then evaluate the integer is formed from the consecutive initial prime product under an explicitly stated sign convention. The scope is broad within that domain but bounded by the need for the integer is formed from the consecutive initial prime product under an explicitly stated sign convention. 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 integer is formed from the consecutive initial prime product under an explicitly stated sign convention 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 number 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 number. Euclid number 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 ordered primes, nth primorial, plus-or-minus-one convention, integer factorization, and divisibility relations. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the integer is formed from the consecutive initial prime product under an explicitly stated sign convention independently of one notation or implementation. This step prevents the canonical example from becoming the definition.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the ordered primes, nth primorial, plus-or-minus-one convention, integer factorization, and divisibility relations, Adding or subtracting one makes the number coprime to every prime included in the primorial, though the resulting number need not itself be prime., and type the carrier, state every parameter and convention in the definition, test that the integer is formed from the consecutive initial prime product under an explicitly stated sign convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Euclid number Domain-specific

Parents (1) — outgoing typed relations

Euclid number is a kind of Function (Mapping) Prime

What makes it its own entry: Euclid's proof motivates the construction but does not assert that every Euclid number is prime, and products over arbitrary prime sets are broader Euclid-style numbers.

Hierarchy path (1) — routes to 1 parentless root

Euclid number → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Euclid number sits in a crowded region of the domain-specific corpus (14^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

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