Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integer Representations and Trajectories of the 3x+1 Problem   This paper studies certain trajectories of the Collatz function. I show that
if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in
\mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation
$$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ 3^{k+1}$$ where
$0\le a_0 \le a_1 \le \cdot \cdot \cdot \le a_{k+1}$. As a consequence, in
order to prove Collatz Conjecture I illustrate that it is sufficient to prove
$n\sim 3n+2$ for any odd $n\in \mathbb{N}\setminus 2^{\mathbb{N}} $. This is
the main result of the paper.

32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn