3,035 research outputs found
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
Heights and quadratic forms: on Cassels' theorem and its generalizations
In this survey paper, we discuss the classical Cassels' theorem on existence
of small-height zeros of quadratic forms over Q and its many extensions, to
different fields and rings, as well as to more general situations, such as
existence of totally isotropic small-height subspaces. We also discuss related
recent results on effective structural theorems for quadratic spaces, as well
as Cassels'-type theorems for small-height zeros of quadratic forms with
additional conditions. We conclude with a selection of open problems.Comment: 16 pages; to appear in the proceedings of the BIRS workshop on
"Diophantine methods, lattices, and arithmetic theory of quadratic forms", to
be published in the AMS Contemporary Mathematics serie
On the Quantitative Subspace Theorem
In this survey we give an overview of recent developments on the Quantitative
Subspace Theorem. In particular, we discuss a new upper bound for the number of
subspaces containing the "large" solutions, obtained jointly with Roberto
Ferretti, and sketch the proof of the latter. Further, we prove a new gap
principle to handle the "small" solutions in the system of inequalities
considered in the Subspace Theorem. Finally, we go into the refinement of the
Subspace Theorem by Faltings and Wuestholz, which states that the system of
inequalities considered has only finitely many solutions outside some
effectively determinable proper linear subspace of the ambient solution space.
Estimating the number of these solutions is still an open problem. We give some
motivation that this problem is very hard.Comment: 26 page
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