Euler pentagonal number theorem
E697753
The Euler pentagonal number theorem is a fundamental result in number theory and combinatorics that gives a remarkable infinite product expansion for the generating function of partition numbers, involving exponents given by generalized pentagonal numbers.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Euler pentagonal number theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871641 — 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: Euler pentagonal number theorem Context triple: [Jacobi triple product, implies, Euler pentagonal number theorem]
-
A.
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.
-
B.
Jacobi’s four-square theorem
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.
-
C.
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.
-
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.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- 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: Euler pentagonal number theorem Target entity description: The Euler pentagonal number theorem is a fundamental result in number theory and combinatorics that gives a remarkable infinite product expansion for the generating function of partition numbers, involving exponents given by generalized pentagonal numbers.
-
A.
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.
-
B.
Jacobi’s four-square theorem
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.
-
C.
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.
-
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.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in combinatorics ⓘ result in number theory ⓘ |
| appliesTo |
analytic functions in the unit disk via q-series
ⓘ
formal power series in one variable ⓘ |
| characterizes | coefficients in the expansion of ∏_{n≥1}(1−x^n) ⓘ |
| connectsTo |
Euler function φ(q)=∏_{n≥1}(1−q^n)
ⓘ
modular forms ⓘ q-series ⓘ |
| describes | generating function of partition numbers ⓘ |
| expresses | alternating sum over generalized pentagonal exponents ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ partition theory ⓘ |
| gives | infinite product expansion for the partition generating function ⓘ |
| hasConsequence |
explicit formula for coefficients using generalized pentagonal numbers
ⓘ
sign pattern of coefficients in the product ∏_{n≥1}(1−x^n) ⓘ |
| hasDomain | complex variable q with |q|<1 in analytic context ⓘ |
| hasProofTechnique |
combinatorial arguments
ⓘ
generating functions ⓘ manipulation of infinite products ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| implies | recurrence relations for the partition function p(n) ⓘ |
| introducedBy | Leonhard Euler ⓘ |
| involves | generalized pentagonal numbers ⓘ |
| isCitedIn |
standard texts on combinatorial generating functions
ⓘ
standard texts on partition theory ⓘ |
| isFundamentalFor |
the study of integer partitions
ⓘ
the theory of q-series identities ⓘ |
| isPartOf | the theory of integer partitions ⓘ |
| isUsedIn |
asymptotic analysis of partition numbers
ⓘ
combinatorial proofs about partitions ⓘ proofs of identities in q-series ⓘ |
| isUsedTo | derive recurrence for p(n) involving p(n−k(3k−1)/2) ⓘ |
| namedAfter | Leonhard Euler ⓘ |
| relatedTo |
Euler’s recurrence for the partition function
ⓘ
Jacobi triple product identity ⓘ
linked to:
Jacobi triple product
Rogers–Ramanujan identities ⓘ |
| relates | infinite product expansions and power series expansions ⓘ |
| states | the infinite product ∏_{n≥1}(1−x^n) equals ∑_{k=−∞}^{∞}(−1)^k x^{k(3k−1)/2} ⓘ |
| topic | generalized pentagonal numbers sequence 1,2,5,7,12,15,… ⓘ |
| uses | generalized pentagonal numbers k(3k−1)/2 for integers k ⓘ |
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: Euler pentagonal number theorem Description of subject: The Euler pentagonal number theorem is a fundamental result in number theory and combinatorics that gives a remarkable infinite product expansion for the generating function of partition numbers, involving exponents given by generalized pentagonal numbers.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.