Fermat number
E146191
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Fermat prime | 2 |
| Fermat number canonical | 1 |
| Fermat numbers | 1 |
| Fermat prime is a Fermat number that is prime | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1281484 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fermat number Context triple: [Pierre de Fermat, notableWork, Fermat number]
-
A.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fermat number
Target entity description: A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
-
A.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
class of integers
ⓘ
concept in number theory ⓘ |
| application | construction of regular polygons ⓘ |
| conjecturedProperty | no further Fermat primes beyond n = 4 are known ⓘ |
| constructibilityCriterion | regular n‑gon is constructible if n is product of a power of 2 and distinct Fermat primes ⓘ |
| coprimeProperty | any two distinct Fermat numbers are coprime ⓘ |
| definitionOfRelatedConcept |
Fermat number
self-linksurface differs
ⓘ
surface form:
Fermat prime is a Fermat number that is prime
|
| domainOfVariable | non‑negative integers ⓘ |
| fifthTerm | F_4 = 65537 ⓘ |
| firstTerm | F_0 = 3 ⓘ |
| fourthTerm | F_3 = 257 ⓘ |
| greaterThan | 1 ⓘ |
| growthRate | doubly exponential in n ⓘ |
| hasGeneralForm | F_n = 2^{2^n} + 1 ⓘ |
| hasProperty |
important in primality testing research
ⓘ
pairwise relatively prime ⓘ rapidly growing sequence ⓘ |
| hasSequenceName |
Fermat number
self-linksurface differs
ⓘ
surface form:
Fermat numbers
|
| hasVariable | n ⓘ |
| isInteger | true ⓘ |
| isOdd | true ⓘ |
| isPrimeForIndex |
n = 0
ⓘ
n = 1 ⓘ n = 2 ⓘ n = 3 ⓘ n = 4 ⓘ |
| knownCompositeForIndex |
n = 10
ⓘ
n = 11 ⓘ n = 12 ⓘ n = 13 ⓘ n = 14 ⓘ n = 15 ⓘ n = 5 ⓘ n = 6 ⓘ n = 7 ⓘ n = 8 ⓘ n = 9 ⓘ |
| modularProperty | F_n ≡ 2 (mod F_0 F_1 … F_{n-1}) ⓘ |
| namedAfter | Pierre de Fermat ⓘ |
| OEISID | A000215 ⓘ |
| openProblem |
whether infinitely many Fermat primes exist
ⓘ
whether infinitely many composite Fermat numbers exist ⓘ |
| productPlusTwoIdentity | F_0 F_1 … F_{n-1} = F_n - 2 ⓘ |
| relatedTo |
Fermat number
self-linksurface differs
ⓘ
surface form:
Fermat prime
constructible polygon ⓘ |
| secondTerm | F_1 = 5 ⓘ |
| sequenceIndexNotation | F_n ⓘ |
| studiedIn |
algebraic number theory
ⓘ
elementary number theory ⓘ |
| thirdTerm | F_2 = 17 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fermat number
Description of subject: A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fermat prime
this entity surface form:
Fermat prime
this entity surface form:
Fermat prime is a Fermat number that is prime
this entity surface form:
Fermat numbers