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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Ramanujan’s congruences for partition function | 1 |
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 NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ramanujan’s congruences for partition function
subject surface form:
Dyson’s transform