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The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic , now known as Ramanujan's congruences .
It is seen to have dimension 0 only in the cases where ℓ = 5, 7 or 11 and since the partition function can be written as a linear combination of these functions [4] this can be considered a formalization and proof of Ramanujan's observation.
Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n {\displaystyle n} ends in the digit 4 or 9, the number of partitions of n {\displaystyle n} will be divisible by 5.
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions.The identities were first discovered and proved by Leonard James Rogers (), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913.
Let n be a non-negative integer and let p(n) denote the number of partitions of n (p(0) is defined to be 1).Srinivasa Ramanujan in a paper [3] published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences.
It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
In 2000, Ono derived a theory of Ramanujan congruences for the partition function with all prime moduli greater than 3. His paper was published in the Annals of Mathematics. [8] In a joint work with Jan Bruinier, Ono discovered a finite algebraic formula for computing partition numbers. [9]