Jacobi’s four-square theorem
E182757
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi’s four-square theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615226 — 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: Jacobi’s four-square theorem Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi’s four-square theorem]
-
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'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.
-
C.
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.
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D.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
-
E.
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.
- 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: Jacobi’s four-square theorem Target entity description: Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
-
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'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.
-
C.
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.
-
D.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in additive number theory
ⓘ
theorem in number theory ⓘ |
| appearsIn |
Hardy and Wright, An Introduction to the Theory of Numbers
ⓘ
linked to:
An Introduction to the Theory of Numbers
Serre, A Course in Arithmetic ⓘ
linked to:
A Course in Arithmetic
classical texts on number theory ⓘ |
| appliesTo | positive integers ⓘ |
| classification | exact formula for representation numbers ⓘ |
| counts | ordered representations of n as x^2 + y^2 + z^2 + t^2 with integer variables ⓘ |
| countsZeroRepresentation | includes representations with zero coordinates ⓘ |
| describes | number of representations of an integer as a sum of four squares ⓘ |
| domainOfFormula | n ∈ ℕ ⓘ |
| excludes |
identifying sign-equivalent representations
ⓘ
unordered representations ⓘ |
| field |
additive number theory
ⓘ
number theory ⓘ |
| formulaType | multiplicative in n ⓘ |
| generalizationOf | earlier results on sums of two squares ⓘ |
| givesFormulaFor | r_4(n) ⓘ |
| givesGeneratingFunction | (θ_3(q))^4 = 1 + ∑_{n≥1} r_4(n) q^n ⓘ |
| hasConsequence | explicit formula for r_4(p^k) for prime powers ⓘ |
| implies |
Lagrange’s four-square theorem
ⓘ
linked to:
Lagrange's four-square theorem
every positive integer has at least one representation as a sum of four squares ⓘ |
| includesSignAndOrder | yes ⓘ |
| mathematicalSubjectClassification |
11E25
ⓘ
11F27 ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| prover | Carl Gustav Jacob Jacobi ⓘ |
| refines |
Lagrange’s four-square theorem
ⓘ
linked to:
Lagrange's four-square theorem
|
| relatedTo |
Jacobi triple product
ⓘ
Lagrange’s four-square theorem ⓘ
linked to:
Lagrange's four-square theorem
modular form of weight 2 ⓘ theta function θ_3(q) ⓘ |
| representationType | ordered quadruples (x,y,z,t) ⓘ |
| statesThat |
r_4(n) = 8 * (σ(n) - 4σ(n/4)) with σ(n/4)=0 if 4∤n
ⓘ
r_4(n) = 8 * sum_{d|n, 4∤d} d ⓘ |
| symbol | r_4(n) ⓘ |
| topic |
representation of integers by quadratic forms
ⓘ
sum of four squares ⓘ |
| usedIn |
analytic number theory
ⓘ
theory of modular forms ⓘ theory of quadratic forms ⓘ |
| uses |
divisor function
ⓘ
modular forms ⓘ theta functions ⓘ |
| variableCondition | x, y, z, t ∈ ℤ ⓘ |
| yearProved | 19th century ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Jacobi’s four-square theorem Description of subject: Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.