Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two terms with known prime divisors adding to a power: REVISED   Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with
$c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$,
$z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely
those in $R$. We consider $N$, the number of solutions of such an equation.
Given a solution $(X, Y, z)$, let $D$ be the least positive integer such that
$(XY/D)^{1/2}$ is an integer. Further, let $\omega$ be the number of distinct
primes dividing $c$. Standard elementary approaches use an upper bound of $2^n$
for the number of possible $D$, and an upper bound of $2^{\omega-1}$ for the
number of ideal factorizations of $c$ in the field $\mathbb{Q}(\sqrt{-D})$
which can correspond (in a standard designated way) to a solution in which
$(XY/D)^{1/2} \in \mathbb{Z}$, and obtain $N \le 2^{n+\omega-1}$. Here we
improve this by finding an inverse proportionality relationship between a bound
on the number of $D$ which can occur in solutions and a bound (independent of
$D$) on the number of ideal factorizations of $c$ which can correspond to
solutions for a given $D$. We obtain $N \le 2^{n-1}+1$. The bound is precise
for $n<4$: there are several cases with exactly $2^{n-1} + 1$ solutions.

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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: physics