Ramanujan partition congruences
E94841
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Ramanujan’s congruences for partition function | 2 |
| Ramanujan partition congruences canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T795892 — 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: Ramanujan partition congruences Context triple: [Dyson’s transform, relatedTo, Ramanujan partition congruences]
-
A.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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B.
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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C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
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E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- 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: Ramanujan partition congruences Target entity description: 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.
-
A.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
B.
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.
-
C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
number theory result ⓘ partition congruence ⓘ |
| appliesTo | partition function p(n) ⓘ |
| concernsFunction | p(n) ⓘ |
| describes |
arithmetic progressions with special partition divisibility
ⓘ
divisibility patterns in partition numbers ⓘ modular properties of the partition function ⓘ |
| discoverer | Srinivasa Ramanujan ⓘ |
| exampleOf | congruence in combinatorial number theory ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ |
| firstCongruence | p(5k+4) ≡ 0 (mod 5) ⓘ |
| firstCongruenceModulus | 5 ⓘ |
| generalizedBy |
Atkin congruences
ⓘ
Ono’s partition congruences ⓘ |
| hasPrimeModulus |
11
ⓘ
5 ⓘ 7 ⓘ |
| importance | foundational in the arithmetic theory of partitions ⓘ |
| inspired |
development of the theory of modular forms
ⓘ
study of congruences for partition functions modulo primes ⓘ |
| involves |
arithmetic progressions
ⓘ
modular arithmetic ⓘ prime moduli ⓘ |
| laterProvedUsing |
Hecke theory
ⓘ
modular forms theory ⓘ p-adic modular forms ⓘ |
| mainSubject |
integer partitions
ⓘ
partition function ⓘ |
| notableFor |
simple arithmetic progression patterns
ⓘ
unexpected divisibility of partition numbers ⓘ |
| patternType | linear congruences for p(n) ⓘ |
| property | show that certain partition numbers are always divisible by a given prime ⓘ |
| provedBy | Srinivasa Ramanujan ⓘ |
| relatedTo |
Hecke operators
ⓘ
Ramanujan tau function ⓘ modular equations ⓘ modular forms ⓘ q-series ⓘ |
| secondCongruence | p(7k+5) ≡ 0 (mod 7) ⓘ |
| secondCongruenceModulus | 7 ⓘ |
| statedIn | paper on highly composite numbers and partitions ⓘ |
| status | proved ⓘ |
| thirdCongruence | p(11k+6) ≡ 0 (mod 11) ⓘ |
| thirdCongruenceModulus | 11 ⓘ |
| yearProposed | 1919 ⓘ |
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: Ramanujan partition congruences Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Dyson’s transform
this entity surface form:
Ramanujan’s congruences for partition function
this entity surface form:
Ramanujan’s congruences for partition function