255 research outputs found
Smooth values of some quadratic polynomials
In this paper, using a method of Luca and the author, we find all values
such that the quadratic polynomials and are
200-smooth and all values such that the quadratic polynomial is
100-smooth.Comment: 12 pages, to appear in Glas. Mat. Ser. II
Descent methods for studying integer points on
We study the integer points on superelliptic and hyperelliptic curves of the
form $n\ge 2, {\rm{deg}}{f}+{\rm{deg}}{g}\ge 4.
Constructions of diagonal quartic and sextic surfaces with infinitely many rational points
In this note we construct several infinite families of diagonal quartic
surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where
with infinitely many rational points and
satisfying the condition . In particular, we present an
infinite family of diagonal quartic surfaces defined over \Q with Picard
number equal to one and possessing infinitely many rational points. Further, we
present some sextic surfaces of type , , , or
, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor
Integral points on a certain family of elliptic curves
The Thue-Siegel method is used to obtain an upper bound for the number of
primitive integral solutions to a family of quartic Thue's inequalities. This
will provide an upper bound for the number of integer points on a family of
elliptic curves with j-invariant equal to 1728
Integral Points for Multi-norm Tori
We construct a finite subgroup of Brauer-Manin obstruction for detecting the
existence of integral points on integral models of principle homogeneous spaces
of multi-norm tori. Several explicit examples are provided.Comment: 25 pages. This version is organized much different from the last on
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