Lagrange's four-square theorem

E156185

result in additive number theory theorem in number theory

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.

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All labels observed (2)

Label	Occurrences
Lagrange’s four-square theorem	5
Lagrange's four-square theorem canonical	4

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.
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
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 for k-th powers ⓘ linked to: Waring's problem 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 theorem ⓘ linked to: Waring's problem 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

Referenced by (9)

Full triples — surface form annotated when it differs from this entity's canonical label.

Joseph-Louis Lagrange → knownFor → Lagrange's four-square theorem ⓘ

Fermat's theorem on sums of two squares → relatedTo → Lagrange's four-square theorem ⓘ

Fermat polygonal number theorem → specialCase → Lagrange’s four-square theorem ⓘ