Euclid number
Let an agent explain Euclid numbers and their relation to Euclid proof, relay known primality results and open problems from number theory references, and distinguish Euclid numbers from primorial primes and Euclid-Mullin sequences.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a Euclid number is.
Definition
Definition.
Definition
Definition.
- What is a Euclid number, and how does it relate to Euclid proof that there are infinitely many primes? definition
- Is the question about Euclid numbers, primorial primes or the Euclid-Mullin sequence? boundary
Examples
Examples.
Examples
Examples.
- What are the first Euclid numbers, and which are prime or composite? measurement
- Which entry fits the specific number? action
Theory Results.
Results
Known results.
Results
Results.
- What is known about the primality and factors of Euclid numbers? provenance
- Which references are standard? provenance
Open
Open problems.
Open
Open.
- Is it known whether infinitely many Euclid numbers are prime or composite? provenance
- Which sources are cited? provenance
Compare Related sequences.
Primorial
Primorials and primorial primes.
Primorial
Primorial.
- How do primorials and primorial primes relate to Euclid numbers? provenance
- Which entry fits primorial? action
Mullin
Euclid-Mullin sequence.
Mullin
Mullin.
- What is the Euclid-Mullin sequence, and how does it differ? provenance
- Which entry fits Euclid-Mullin sequence? action
Context History and teaching.
History
Euclid proof.
History
History.
- How does Euclid proof work, and how is it often misstated? provenance
- Which entry fits Euclid theorem? action
Computation
Computation.
Computation
Computation.
- How are large Euclid numbers tested for primality? action
- Which entry fits primality test? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A number of the form E_n = p_n# + 1, one more than the product of the first n prime numbers, named after Euclid because his proof of the infinitude of primes considers such products, giving 3, 7, 31, 211, 2311, 30031 and so on; the first few Euclid numbers are prime but 30031 is composite, it is unknown whether infinitely many Euclid numbers are prime, and they are related to primorials and to the primorial primes.
Why it exists Filled
Let an agent explain Euclid numbers and their relation to Euclid proof, relay known primality results and open problems from number theory references, and distinguish Euclid numbers from primorial primes and Euclid-Mullin sequences.
Distinguishing features Filled
- Primorial plus one
- Euclid proof link
- Mostly composite
- Open primality question
What robots and AI may and may not do Filled
Must not
- State that every Euclid number is prime.
- Claim that Euclid's proof constructs new primes directly as Euclid numbers.
- Claim new primality results without verifiable computation or a published source.
Only with a human decision
- Announcing a new mathematical result about Euclid numbers to the public.
May
- Compute Euclid numbers and state which are known to be prime or composite, citing sources.
- Explain the role of such products in Euclid's proof of the infinitude of primes.
- Distinguish Euclid numbers from primorial primes and the Euclid-Mullin sequence.
Moral aspects Filled
- The common misreading of Euclid's proof spreads a false lesson to students.
Who is affected
- Students and teachers
- Mathematicians
Owners Filled
Steward
Nobody: a mathematical concept is held in common.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- primorial - plus one
- Euclid - and his proof of infinite primes
- primorial prime - when prime
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.euclid-number
- Wikidata: Q728223 (https://www.wikidata.org/wiki/Q728223)
- OEIS: A006862 Euclid numbers
Direct properties not applicable Not applicable
- first Euclid numbers: 3, 7, 31, 211, 2311, 30031 sequence - OEIS A006862
- first composite Euclid number: 30031 value - 59 times 509
- known prime Euclid numbers: for n = 1 to 5, 11, 75, 171, 172, 384 and more list - per OEIS
Plane XCT: no invented physical properties.
Recognition optional Filled
- Primorial plus one
- E_1 = 3, E_2 = 7, E_3 = 31, E_4 = 211, E_5 = 2311, E_6 = 30031
- Primorial primes include p# - 1 forms; the Euclid-Mullin sequence uses smallest prime factors
- Not a visible object; integers one more than primorials.
Capabilities and actions required Filled
- explain the definition
- relay known results
- describe open problems
- distinguish related sequences
Hazards and failure modes required Filled
- Assuming all Euclid numbers are prime
- Misreading Euclid proof as producing primes directly
- Confusing with primorial primes or the Euclid-Mullin sequence
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- Euclid numbers p_n# + 1
- primorial primes of the form p# plus or minus 1
- Euclid-Mullin sequence
- Kummer numbers p_n# - 1
Sources Filled
- Wikidata item Q728223: Euclid number - identity and sense of the item
- Wikipedia: Euclid number - general description of the item
Open questions
- Should primorial and primorial prime be the primary linked entries?
- How should number theory references be linked?
- How should the record of prime Euclid numbers be kept current?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q728223/spec.json