Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The number of equations c=a+b satisfying the abc-conjecture   We prove that for a positive integer $c$ and any given $\varepsilon$,
$0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive
coprime integers $a$ and $b$, which satisfy the inequality $$c <
R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$
where R(n) is the radical of $n$, is for $c\to\infty$
$$N(c)=(1-\varepsilon)\frac{\phi(c)}{2}+O\Bigl(\frac{\phi(c)}{2}\Bigr).$$ An
analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without
a constant factor) will also be proved.

33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: q-fin