Fermat polygonal number theorem
E147900
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Fermat polygonal number theorem canonical | 1 |
| Fermat’s theorem on polygonal numbers | 1 |
| Gauss’s Eureka theorem on triangular numbers | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1281489 — 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 polygonal number theorem Context triple: [Pierre de Fermat, notableWork, Fermat polygonal number theorem]
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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'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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C.
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.
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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.
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.
- 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 polygonal number theorem Target entity description: 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.
-
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'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.
-
C.
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.
-
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.
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.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem ⓘ |
| alsoKnownAs |
Fermat polygonal number theorem
ⓘ
surface form:
Fermat’s theorem on polygonal numbers
|
| appliesTo |
hexagonal numbers
ⓘ
k-gonal numbers for any integer k ≥ 3 ⓘ pentagonal numbers ⓘ square numbers ⓘ triangular numbers ⓘ |
| assumes | standard arithmetic of the integers ⓘ |
| classification | classical theorem in additive number theory ⓘ |
| concerns |
polygonal numbers
ⓘ
representation of integers as sums of figurate numbers ⓘ |
| dealsWith | finite sums of polygonal numbers ⓘ |
| example |
every positive integer is a sum of five pentagonal numbers
ⓘ
every positive integer is a sum of four square numbers ⓘ every positive integer is a sum of six hexagonal numbers ⓘ every positive integer is a sum of three triangular numbers ⓘ |
| field | number theory ⓘ |
| generalizes |
Fermat polygonal number theorem
self-linksurface differs
ⓘ
surface form:
Gauss’s Eureka theorem on triangular numbers
Lagrange's four-square theorem ⓘ
surface form:
Lagrange’s four-square theorem
|
| historicalClaimBy | Pierre de Fermat ⓘ |
| implies | existence of finite additive bases formed by polygonal numbers of fixed order ⓘ |
| involves | minimal number of s-gonal numbers needed to represent any positive integer ⓘ |
| logicalForm | universal-existential statement about representations of integers ⓘ |
| mathematicalDomain | theory of figurate numbers ⓘ |
| namedAfter | Pierre de Fermat ⓘ |
| orderDependentBound | for each s ≥ 3 there exists a minimal r(s) such that every positive integer is a sum of r(s) s-gonal numbers ⓘ |
| quantifier |
every positive integer
ⓘ
fixed number of polygonal numbers depending on the order ⓘ |
| relatedConcept |
Waring's problem
ⓘ
surface form:
Waring’s problem
sum of polygonal numbers ⓘ sum of powers ⓘ |
| specialCase |
Lagrange's four-square theorem
ⓘ
surface form:
Lagrange’s four-square theorem
theorem that every positive integer is a sum of five pentagonal numbers ⓘ theorem that every positive integer is a sum of four square numbers ⓘ theorem that every positive integer is a sum of six hexagonal numbers ⓘ theorem that every positive integer is a sum of three triangular numbers ⓘ |
| statement | every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order ⓘ |
| statusOfFermatProof | Fermat did not leave a complete proof ⓘ |
| topic | additive number theory ⓘ |
| typeOfResult | existence theorem ⓘ |
| usesConcept |
figurate numbers
ⓘ
k-gonal numbers ⓘ |
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 polygonal number theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fermat’s theorem on polygonal numbers
Fermat polygonal number theorem
→
generalizes
→
Fermat polygonal number theorem
self-linksurface differs
ⓘ
this entity surface form:
Gauss’s Eureka theorem on triangular numbers