Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the remarkable properties of the pentagonal numbers   In this paper Euler considers the properties of the pentagonal numbers, those
numbers of the form $\frac{3n^2 \pm n}{2}$. He recalls that the infinite
product $(1-x)(1-x^2)(1-x^3)...$ expands into an infinite series with exponents
the pentagonal numbers, and tries substituting the roots of this infinite
product into this infinite series. I am not sure what he is doing in some
parts: in particular, he does some complicated calculations about the roots of
unity and sums of them, their squares, reciprocals, etc., and also sums some
divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any
suggestions or corrections about these parts.

32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an