Ono’s partition congruences
E438308
Ono’s partition congruences are modern number-theoretic results that extend Ramanujan’s classical congruences by proving the existence of infinitely many congruence relations for the partition function modulo various primes.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ono’s partition congruences canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4410548 — 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: Ono’s partition congruences Context triple: [Ramanujan partition congruences, generalizedBy, Ono’s partition congruences]
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A.
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.
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B.
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.
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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.
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D.
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.
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E.
Ramanujan’s lost notebook
Ramanujan’s lost notebook is a posthumously discovered collection of Srinivasa Ramanujan’s final mathematical formulas and insights, many of which were decades ahead of their time in number theory and q-series.
- 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: Ono’s partition congruences Target entity description: Ono’s partition congruences are modern number-theoretic results that extend Ramanujan’s classical congruences by proving the existence of infinitely many congruence relations for the partition function modulo various primes.
-
A.
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.
-
B.
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.
-
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.
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.
-
E.
Ramanujan’s lost notebook
Ramanujan’s lost notebook is a posthumously discovered collection of Srinivasa Ramanujan’s final mathematical formulas and insights, many of which were decades ahead of their time in number theory and q-series.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
number-theoretic result
ⓘ
partition congruence ⓘ result in modular forms ⓘ |
| asserts |
existence of infinitely many arithmetic progressions with partition congruences
ⓘ
existence of infinitely many partition congruences modulo a given prime ⓘ |
| concerns |
congruence properties of the partition function
ⓘ
values of the partition function modulo prime powers ⓘ values of the partition function modulo primes ⓘ |
| context |
arithmetic of special values of modular functions
ⓘ
theory of modular forms and q-series ⓘ |
| extends | Ramanujan’s classical partition congruences ⓘ |
| field |
algebraic number theory
ⓘ
analytic number theory ⓘ number theory ⓘ |
| generalizes |
Ramanujan’s congruence p(11n+6) ≡ 0 (mod 11)
ⓘ
Ramanujan’s congruence p(5n+4) ≡ 0 (mod 5) ⓘ Ramanujan’s congruence p(7n+5) ≡ 0 (mod 7) ⓘ |
| hasConsequence | existence of congruences for many other partition-like functions ⓘ |
| implies |
infinitely many congruences for the partition function modulo 11
ⓘ
infinitely many congruences for the partition function modulo 2 ⓘ infinitely many congruences for the partition function modulo 3 ⓘ infinitely many congruences for the partition function modulo 5 ⓘ infinitely many congruences for the partition function modulo 7 ⓘ |
| influenced |
research on congruences for other q-series
ⓘ
subsequent work on partition congruences by Ahlgren ⓘ subsequent work on partition congruences by Boylan ⓘ subsequent work on partition congruences by Mahlburg ⓘ |
| involves |
Chebotarev density theorem techniques
ⓘ
arithmetic progressions of the form An+B ⓘ density results for congruence classes ⓘ modular forms with complex multiplication ⓘ |
| namedAfter | Ken Ono ⓘ |
| provedBy | Ken Ono ⓘ |
| publishedIn | Annals of Mathematics ⓘ |
| relatedTo |
Galois representations
ⓘ
Hecke operators ⓘ Ramanujan’s partition congruences ⓘ
linked to:
Ramanujan partition congruences
modular curves ⓘ modular forms ⓘ p-adic modular forms ⓘ partition function ⓘ |
| uses |
Deligne’s work on Galois representations
ⓘ
Galois representations attached to modular forms ⓘ Hecke eigenforms ⓘ Serre’s theory of modular forms mod p ⓘ properties of modular forms of half-integral weight ⓘ properties of modular forms of integral weight ⓘ |
| yearProved | 2000 ⓘ |
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Subject: Ono’s partition congruences Description of subject: Ono’s partition congruences are modern number-theoretic results that extend Ramanujan’s classical congruences by proving the existence of infinitely many congruence relations for the partition function modulo various primes.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.