Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The second Hardy-Littlewood conjecture is true   The second Hardy-Littlewood conjecture, that $\pi(x)+\pi(y) \geq \pi(x+y)$
for integers $x$ and $y$ with $\min\{x,y\}\geq 2$, was formulated in 1923. It
continues to attract attention to this day, almost 100 years later. In 1975
Udrescu proved that this conjecture holds for $(x,y)$ sufficiently large, but
without an explicit effective bound on the region of validity. We shall revisit
Udrescu's result, modifying it to obtain explicit effective bounds, ultimately
proving that the second Hardy-Littlewood conjecture is in fact unconditionally
true. Furthermore we note that constraints on the prime counting function
imply, (and are implied by), constraints on the location of the primes, and
re-cast Segal's 1962 equivalent reformulation of the second Hardy-Littlewood
conjecture in the more symmetric (and perhaps clearer) form that for integers
$i$ and $j$ with $\min\{i,j\} \geq 2$ one has $p_{i+j-1} \geq p_i + p_j -1$.

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The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
3
The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat