Euler's Pentagonal Numbers Theorem

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Theorem

Consider the infinite product:

$\displaystyle P = \prod_{n \mathop \in \Z_{>0} } \paren {1 - x^n}$

Then $P$ can be expressed as:

$\displaystyle P = \sum_{n \mathop \in \Z_{>0} } \paren {-1}^{\ceiling {n / 2} } x^{GP_n}$

where:

$\ceiling {n / 2}$ denotes the ceiling of $n / 2$
$GP_n$ denotes the $n$th generalized pentagonal number.


That is:

$P = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots$


Corollary 1

Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\map \sigma n$ denote the $\sigma$ function on $n$.


Then:

$\map \sigma n = \displaystyle \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map \sigma {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$


Corollary 2

Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\map p n$ denote the number of partitions on $n$.


Then:

$\map p n = \displaystyle \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map p {n - GP_k} + \sqbrk {\exists k \in \Z: GP_k = n}$


Proof


Historical Note

This result was first noticed by Leonhard Paul Euler, who noticed the pattern on starting to multiply out the given infinite product, but he was initially unable to prove it except by a process of informal induction.

Despite this inability, he was so certain of the pattern that he used it as the basis of the development of further theorems.


Sources