Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the 1-3-5 conjecture and related topics   The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb
N=\{0,1,2,\ldots\}$ can be written as $x^2+y^2+z^2+w^2$ with $w,x,y,z\in\mathbb
N$ such that $x+3y+5z$ is a square. In this paper, via the theory of ternary
quadratic forms and related modular forms, we study the integer version of the
1-3-5 conjecture and related weighted sums of four squares with certain linear
restrictions. Here are two typical results in this paper: (i) There is a finite
set $A$ of positive integers such that any sufficiently large integer not in
the set $\{16^ka:\ a\in A,\ k\in\mathbb N\}$ can be written as
$x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ and $x+3y+5z\in\{4^k:\ k\in\mathbb
N\}$. (ii) Any positive integer can be written as $x^2+y^2+z^2+2w^2$ with
$x,y,z,w\in\mathbb Z$ and $x+y+2z+2w=1$. Also, any sufficiently large integer
can be written as $x^2+y^2+z^2+2w^2$ with $x,y,z,w\in\mathbb Z$ and
$x+2y+3z=1$.

12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
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The categories of this paper are: econ
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org