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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 number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.)
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.
then the partition λ is called a strict partition of n. The integers λ k, λ k − 1, ..., λ 1 are the parts of the partition. The number of parts in the partition λ is k and the largest part in the partition is λ k. The rank of the partition λ (whether ordinary or strict) is defined as λ k − k. [1]
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
The first belongs to a family of formulas which were rigorously proven by the Chudnovsky brothers in 1989 [11] and later used to calculate 10 trillion digits of π in 2011. [12] The second formula, and the ones for higher levels, was established by H.H. Chan and S. Cooper in 2012.