Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small gaps between almost primes, the parity problem, and some
  conjectures of Erd\H{o}s on consecutive integers II   This paper is intended as a sequel to a paper arXiv:0803.2636 written by four
of the coauthors here. In the paper, they proved a stronger form of the
Erd\H{o}s-Mirksy conjecture which states that there are infinitely many
positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number
of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984,
but the method did not reveal the nature of the set of values $d(x)$ for such
$x$. In particular, one could not conclude that there was any particular value
$A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper
arXiv:0803.2636, the authors showed that there are infinitely many positive
integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$,
so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e.,
$x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was
done for shifts $n$ which are either even or not divisible by the product of a
pair of twin primes. The goal of this paper is to give simple proofs of results
on exponent patterns for an arbitrary shift $n$.

33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn