Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Binary quadratic forms and sums of powers of integers   In this methodological paper, we first review the classic cubic Diophantine
equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions
$q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form.
Next we turn our attention to the familiar sums of powers of the first $n$
positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and express the squares
$S_k^2$, $S_m^2$, and the product $S_k S_m$ as a linear combination of power
sums. These expressions, along with the above quadratic-form solution for the
cubic equation, allows one to generate an infinite number of relations of the
form $Q_1^3 + Q_2^3 + Q_3^3 = Q_4^3$, with each $Q_i$ being a linear
combination of power sums. Also, we briefly consider the quadratic Diophantine
equations $a^2 + b^2 + c^2 = d^2$ and $a^2 + b^2 = c^2$, and give a family of
corresponding solutions $Q_1^2 + Q_2^2 + Q_3^2 = Q_4^2$ and $Q_1^2 + Q_2^2 =
Q_3^2$ in terms of sums of powers of integers.

14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
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The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn