Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Upper Bounds with $ABC=2^m p^n$ and $ABC= 2^m p^n q^r$ with $p$ and
  $q$ as Mersenne or Fermat Primes   Per the $ABC$ conjecture, for every $\varepsilon > 0$, there exist finitely
many triples $\{A, B, C\}$, satisfying $A + B = C, gcd(A,B) = 1, B > A \geq 1$,
such that $C > rad(ABC)^{1 + \varepsilon}$, where $rad(ABC)$ is the product of
all distinct primes constituting $ABC$. It is shown (possibly known) that for
any $\varepsilon > 0$, $rad(ABC)^{1 + \varepsilon} > C$ holds for $ABC$ of the
form $2^mp^n$ such that $2^m + \mu = p^n$, where $m, n$ are positive integers,
$\mu = \pm 1$, and $p$ is an odd prime. Condition $2^m + \mu = p^n$ requires
$p$ to be either a Mersenne or Fermat prime. For $ABC = 2^mp^nq^r$, $p$ and $q$
either Fermat or Mersenne primes but distinct, and $m,n,r$ positive integers,
such that any diophantine relation, $\mbox{(i)}\; 2^m + \mu = p^nq^r,
\mbox{(ii)} \; 2^mp^n + \mu = q^r \mbox{ or } \mbox{(iii)} \; p^n + \mu q^r =
2^m$, holds, using results in literature and mostly elementary methods, the
number of triples $\{A, B, C \}$ are shown to be finite, except for $\{2^{y+1},
2^{2y}+1, (2^y+1)^2\}$, a solution to (iii) with $y$ such that $2^y+1$ and
$2^{2y}+1$ are primes, in which case $rad(ABC)^{1 + \varepsilon} > C$ holds for
any $\varepsilon > 0$.

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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph