Search results
Results from the WOW.Com Content Network
For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly.
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.
4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1. The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part.
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.
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 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later other congruences of this type were discovered, for numbers and for Tau-functions.
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.
Srinivasa Ramanujan (picture) was bedridden when he developed the idea of taxicab numbers, according to an anecdote from G. H. Hardy.. In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. [1]
1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically. 1729 is also known as Ramanujan number or Hardy–Ramanujan number, named after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan who was ill in a hospital.