Lagrange's four-square theorem
E156185
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lagrange’s four-square theorem | 5 |
| Lagrange's four-square theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358586 — 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: Lagrange's four-square theorem Context triple: [Joseph-Louis Lagrange, knownFor, Lagrange's four-square theorem]
-
A.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
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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.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
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D.
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.
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E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- 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: Lagrange's four-square theorem Target entity description: 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.
-
A.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect 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.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
D.
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.
-
E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in additive number theory
ⓘ
theorem in number theory ⓘ |
| allowsRepresentationAs | sum of four squares of integers ⓘ |
| appliesTo | natural numbers ⓘ |
| category | Diophantine equation result ⓘ |
| concerns | representation of integers as sums of squares ⓘ |
| doesNotRequire | uniqueness of representation ⓘ |
| equivalentTo | closure of nonnegative integers under four-square addition formula ⓘ |
| exampleRepresentation |
15 = 9 + 4 + 1 + 1
ⓘ
23 = 16 + 4 + 1 + 2 ⓘ 7 = 4 + 1 + 1 + 1 ⓘ |
| field | number theory ⓘ |
| guarantees | existence of a four-square representation for each natural number ⓘ |
| hasGeneralization |
Waring's problem
ⓘ
surface form:
Waring's problem for k-th powers
sum of k squares theorems ⓘ |
| historicalAttribution | often attributed to Fermat as a conjecture ⓘ |
| holdsFor | 0 as 0^2 + 0^2 + 0^2 + 0^2 ⓘ |
| implies |
every nonnegative integer is a sum of four squares
ⓘ
every positive integer is a norm of a quaternion over the integers ⓘ |
| involves |
integer squares
ⓘ
nonnegative integers ⓘ |
| isSharpBound | 4 is best possible uniform bound for squares ⓘ |
| isTaughtIn |
courses on Diophantine equations
ⓘ
undergraduate number theory courses ⓘ |
| minimalNumberOfSquaresGuaranteed | 4 ⓘ |
| namedAfter | Joseph-Louis Lagrange ⓘ |
| precededBy | results of Fermat on sums of squares ⓘ |
| provedBy | Joseph-Louis Lagrange ⓘ |
| relatedTo |
Hurwitz quaternions
ⓘ
Legendre's three-square theorem ⓘ Waring's problem ⓘ
surface form:
Waring's theorem
modular forms ⓘ quadratic forms ⓘ quaternions ⓘ sum of squares problem ⓘ sum of two squares theorem ⓘ theta functions ⓘ |
| specialCaseOf | Waring's problem ⓘ |
| statement | Every natural number can be expressed as the sum of four integer squares ⓘ |
| usedIn |
additive number theory
ⓘ
analytic number theory ⓘ geometry of numbers ⓘ theory of quadratic forms ⓘ |
| usesInProof |
composition of sums of four squares
ⓘ
properties of quadratic forms ⓘ |
| yearProved | 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: Lagrange's four-square theorem Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lagrange’s four-square theorem
this entity surface form:
Lagrange’s four-square theorem
this entity surface form:
Lagrange’s four-square theorem
this entity surface form:
Lagrange’s four-square theorem
this entity surface form:
Lagrange’s four-square theorem