Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some experiments with Ramanujan-Nagell type Diophantine equations   Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has
exactly six solutions in positive integers. Motivated by this result we are
interested in constructions of Diophantine equations of Ramanujan-Nagell type
$x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not
necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular,
we prove that for each $k$ there exists an infinite set $\cal{S}$ containing
pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have
$\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at
least four solutions in positive integers. Moreover, we construct several
Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five
solutions in non-negative integers. %For example the equation $y^2=130\cdot
3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also
find new examples of equations $x^2=A2^{n}+B$ having six solutions in positive
integers, e.g. the following Diophantine equations has exactly six solutions:
\begin{equation*} \begin{array}{ll} x^2= 57\cdot 2^{n}+117440512 & n=0, 14, 16,
20, 24, 25, x^2= 165\cdot 2^{n}+26404 & n=0, 5, 7, 8, 10, 12. \end{array}
\end{equation*} Moreover, based on an extensive numerical calculations we state
several conjectures on the number of solutions of certain parametric families
of the Diophantine equations of Ramanujan-Nagell type.

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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: econ