Waring's problem
E451523
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Waring's problem canonical | 3 |
| Waring's problem for k-th powers | 1 |
| Waring's problem for primes | 1 |
| Waring's theorem | 1 |
| Waring–Goldbach problem | 1 |
| Waring’s problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552364 — 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: Waring's problem Context triple: [Hardy–Littlewood circle method, appliedTo, Waring's problem]
-
A.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
-
B.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
C.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- 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: Waring's problem Target entity description: Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
-
A.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
-
B.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
C.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- F. None of above. chosen
Statements (53)
| Predicate | Object |
|---|---|
| instanceOf |
additive number theory problem
ⓘ
number theory problem ⓘ |
| asksFor | minimal number of k-th powers needed to represent any natural number ⓘ |
| countryOfOrigin | United Kingdom ⓘ |
| field |
additive number theory
ⓘ
number theory ⓘ |
| focusesOn |
bounds on number of summands
ⓘ
fixed exponents ⓘ |
| hasConjecture |
exact values of G(k) for all k
ⓘ
exact values of g(k) for all k ⓘ |
| hasGeneralForm | for every k ≥ 2 there exists g(k) such that every natural number is a sum of at most g(k) k-th powers ⓘ |
| hasInvariant |
G(k)
ⓘ
g(k) ⓘ |
| hasParameter | exponent k ⓘ |
| hasVariant |
G(k) problem
ⓘ
Waring's problem for primes ⓘ
linked to:
Waring's problem
Waring–Goldbach problem ⓘ
linked to:
Waring's problem
asymptotic Waring's problem ⓘ g(k) problem ⓘ |
| influenced |
development of additive number theory
ⓘ
methods in analytic number theory ⓘ |
| involvesConcept |
existence of finite bounds
ⓘ
k-th powers of natural numbers ⓘ natural numbers ⓘ upper bounds ⓘ |
| involvesFunction | g(k) ⓘ |
| knownValueOfG(k) |
G(2) = 4
ⓘ
G(3) = 9 ⓘ G(4) = 16 ⓘ G(5) = 37 ⓘ |
| knownValueOfg(k) |
g(2) = 4
ⓘ
g(3) = 9 ⓘ g(4) = 19 ⓘ g(5) = 37 ⓘ |
| mainSubject | representation of natural numbers as sums of powers ⓘ |
| namedAfter | Edward Waring ⓘ |
| originalFormulationLanguage | Latin ⓘ |
| proposer | Edward Waring ⓘ |
| relatedTo |
Hilbert's basis theorem (historical context)
ⓘ
linked to:
Hilbert basis theorem
Hilbert's solution of Waring's problem ⓘ Lagrange's four-square theorem ⓘ |
| solutionMethod | non-constructive existence proof ⓘ |
| solutionYear | 1909 ⓘ |
| solvedBy | David Hilbert ⓘ |
| specialCase |
cubes
ⓘ
fourth powers ⓘ squares ⓘ |
| statedInWork | Meditationes Algebraicae ⓘ |
| studiedBy |
G. H. Hardy
ⓘ
Klaus Roth ⓘ R. C. Vaughan ⓘ Srinivasa Ramanujan ⓘ |
| yearProposed | 1770 ⓘ |
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: Waring's problem Description of subject: Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Waring's problem
linked to: Waring's problem
linked to: Waring's problem
linked to: Waring's problem
linked to: Waring's problem