Euclid number¶
An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.
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.
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. 2. 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) — more general patterns this builds on
-
Euclid number is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Euclid number → Function (Mapping)
Neighborhood in Abstraction Space¶
Euclid number sits in a crowded region of the domain-specific corpus (24th 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
- Euler's totient function — 0.92
- Euclid–Mullin sequence — 0.91
- Supernatural number — 0.91
- Prime triplet — 0.91
- Dedekind psi function — 0.91
Computed from structural-signature embeddings · 2026-09-08