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This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. Considering p ( 1000 ) {\displaystyle p(1000)} , the asymptotic formula gives about 2.4402 × 10 31 {\displaystyle 2.4402\times 10^{31}} , reasonably close to the exact answer given above (1.415% larger than the true value).
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 mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: (+) (), (+) (), (+) ().In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy [1] states that the normal order of the number () of distinct prime factors of a number is . Roughly speaking, this means that most numbers have about this number of distinct prime factors.
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.
Since σ k (n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences [35] for the functions. See Ramanujan tau function for some examples. Extend the domain of the partition function by setting p(0) = 1.
Among the 22 partitions of the number 8, there are 6 that contain only odd parts: 7 + 1; 5 + 3; 5 + 1 + 1 + 1; 3 + 3 + 1 + 1; 3 + 1 + 1 + 1 + 1 + 1; 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1; Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts. If we count the ...
The following notations are used to specify how many partitions have a given rank. Let n, q be a positive integers and m be any integer. The total number of partitions of n is denoted by p(n). The number of partitions of n with rank m is denoted by N(m, n). The number of partitions of n with rank congruent to m modulo q is denoted by N(m, q, n).