Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Universal sums of generalized pentagonal numbers   For an integer $x$, an integer of the form $P_5(x)=\frac{3x^2-x}2$ is called
a generalized pentagonal number. For positive integers
$\alpha_1,\dots,\alpha_k$, a sum
$\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=\alpha_1P_5(x_1)+\alpha_2P_5(x_2)+\cdots+\alpha_kP_5(x_k)$
of generalized pentagonal numbers is called universal if
$\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=N$ has an integer solution
$(x_1,x_2,\dots,x_k) \in \mathbb Z^k$ for any non-negative integer $N$. In this
article, we prove that there are exactly $234$ proper universal sums of
generalized pentagonal numbers. Furthermore, the "pentagonal theorem of $109$"
is proved, which states that an arbitrary sum
$\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)$ is universal if and only if
it represents the integers $1, 3, 8, 9, 11, 18, 19, 25, 27, 43, 98$, and $109$.

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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn