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Some experiments with Ramanujan-Nagell type Diophantine equations
Stiller proved that the Diophantine equation has
exactly six solutions in positive integers. Motivated by this result we are
interested in constructions of Diophantine equations of Ramanujan-Nagell type
with many solutions. Here, (thus are not
necessarily positive) and are given integers. In particular,
we prove that for each there exists an infinite set containing
pairs of integers such that for each we have
is square-free and the Diophantine equation has at
least four solutions in positive integers. Moreover, we construct several
Diophantine equations of the form with , each containing five
solutions in non-negative integers. %For example the equation has exactly five solutions with . We also
find new examples of equations 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.Comment: 14 pages, to appear in Galsnik Matematick
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