Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Primes in intervals of bounded length   The Twin Prime conjecture states that there are infinitely many pairs of
distinct primes which differ by $2$. Until recently this conjecture had seemed
to be far out of reach with current techniques. However, in April 2013, Yitang
Zhang proved the existence of a finite bound $B$ such that there are infinitely
many pairs of distinct primes which differ by no more than $B$. This is a
massive breakthrough, making the twin prime conjecture look highly plausible,
and the techniques developed help us to better understand other delicate
questions about prime numbers that had previously seemed intractable. Zhang
even showed that one can take $B = 70000000$. Moreover, a co-operative team,
\emph{polymath8}, collaborating only on-line, had been able to lower the value
of $B$ to ${4680}$. They had not only been more careful in several difficult
arguments in Zhang's original paper, they had also developed Zhang's techniques
to be both more powerful and to allow a much simpler proof (and forms the basis
for the proof presented herein). In November 2013, inspired by Zhang's
extraordinary breakthrough, James Maynard dramatically slashed this bound to
$600$, by a substantially easier method. Both Maynard, and Terry Tao who had
independently developed the same idea, were able to extend their proofs to show
that for any given integer $m\geq 1$ there exists a bound $B_m$ such that there
are infinitely many intervals of length $B_m$ containing at least $m$ distinct
primes. We will also prove this much stronger result herein, even showing that
one can take $B_m=e^{8m+5}$.

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The categories of this paper are: eess
4
The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc