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Concept
Partition function (number theory)
Summary
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. 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. It grows as an exponential function of the square root of its argument. 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. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.
For a positive integer n, p(n) is the number of distinct ways of representing n as a sum of positive integers. For the purposes of this definition, the order of the terms in the sum is irrelevant: two sums with the same terms in a different order are not considered to be distinct.
By convention p(0) = 1, as there is one way (the empty sum) of representing zero as a sum of positive integers. Furthermore p(n) = 0 when n is negative.
The first few values of the partition function, starting with p(0) = 1, are:
Some exact values of p(n) for larger values of n include:
the largest known prime number among the values of p(n) is p(1289844341), with 40,000 decimal digits. Until March 2022, this was also the largest prime that has been proved using elliptic curve primality proving.
Pentagonal number theorem
The generating function for p(n) is given by
The equality between the products on the first and second lines of this formula
is obtained by expanding each factor into the geometric series
To see that the expanded product equals the sum on the first line,
apply the distributive law to the product.
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