View source for Partition function (number theory)

{{MSC|11P}}

A partition of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The partition function $p(n)$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0 \pmod 11$, and others. He also found the asymptotic relation
$$
p(n) \sim \frac{e^{K \sqrt{n}}}{4n\sqrt{3}}\ \ \text{as}\ \ n \rightarrow \infty \ ,
$$
where $K = \pi\sqrt{2/3}$. Later this was completed to an exact series expansion by H. Rademacher (see [[#References|[a2]]]).

One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [[#References|[a3]]]). See also [[Additive number theory]]; [[Additive problems]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy; E. M. Wright; ''An Introduction to the Theory of Numbers'' Oxford University Press (2008) ISBN 0-19-921986-5</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Tom M. Apostol; ''Modular functions and Dirichlet Series in Number Theory'' Graduate Texts in Mathematics '''41''' Springer-Verlag (1990) ISBN 0-387-97127-0</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G.E. Andrews,   "The theory of partitions" , Addison-Wesley  (1976)</TD></TR>
</table>

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