Fermat polygonal number theorem

E147900

result in number theory 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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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]

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
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’s theorem on polygonal numbers ⓘ linked to: Fermat polygonal number theorem
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	Gauss’s Eureka theorem on triangular numbers ⓘ linked to: Fermat polygonal number theorem Lagrange’s four-square theorem ⓘ linked to: 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 ⓘ linked to: Waring's problem sum of polygonal numbers ⓘ sum of powers ⓘ
specialCase	Lagrange’s four-square theorem ⓘ linked to: 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

Referenced by (3)

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

Pierre de Fermat → notableWork → Fermat polygonal number theorem ⓘ

Fermat polygonal number theorem → alsoKnownAs → Fermat’s theorem on polygonal numbers ⓘ

linked to: Fermat polygonal number theorem

Fermat polygonal number theorem → generalizes → Gauss’s Eureka theorem on triangular numbers ⓘ