2,502 research outputs found
On certain arithmetic properties of Stern polynomials
We prove several theorems concerning arithmetic properties of Stern
polynomials defined in the following way: , and . We study also the
sequence e(n)=\op{deg}_{t}B_{n}(t) and give various of its properties.Comment: 20 page
Rational points on certain hyperelliptic curves over finite fields
Let be a field, and . Let us consider the
polynomials , where is a fixed
positive integer. In this paper we show that for each the
hypersurface given by the equation \begin{equation*} S_{k}^{i}:
u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a
rational curve. Using the above and Woestijne's recent results \cite{Woe} we
show how one can construct a rational point different from the point at
infinity on the curves defined over a finite
field, in polynomial time.Comment: Revised version will appear in Bull. Polish Acad. Sci. Mat
Rational points on certain del Pezzo surfaces of degree one
Let and let us consider a del Pezzo
surface of degree one given by the equation . In
this note we prove that if the set of rational points on the curve is infinite, then the set of rational
points on the surface is dense in the Zariski topology.Comment: 8 pages. Published in Glasgow Mathematical Journa
Rational solutions of certain Diophantine equations involving norms
In this note we present some results concerning the unirationality of the
algebraic variety given by the equation \begin{equation*}
N_{K/k}(X_{1}+\alpha X_{2}+\alpha^2 X_{3})=f(t), \end{equation*} where is a
number field, , is a root of an irreducible polynomial
and . We are mainly interested in the case
of pure cubic extensions, i.e. and . We prove that
if \op{deg}f=4 and the variety contains a -rational point
with , then is
-unirational. A similar result is proved for a broad family of quintic
polynomials satisfying some mild conditions (for example this family
contains all irreducible polynomials). Moreover, the unirationality of
(with non-trivial -rational point) is proved for any
polynomial of degree 6 with not equivalent to the polynomial
satisfying the condition , where is the
primitive third root of unity. We are able to prove the same result for an
extension of degree 3 generated by the root of polynomial , provided that with
.Comment: submitte
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
Rational points on certain elliptic surfaces
Let , where f\in\Q[t]\setminus\Q, and let us
assume that \op{deg}f\leq 4. In this paper we prove that if \op{deg}f\leq
3, then there exists a rational base change such that on the
surface there is a non-torsion section. A similar
theorem is valid in case when \op{deg}f=4 and there exists t_{0}\in\Q such
that infinitely many rational points lie on the curve
. In particular, we prove that if \op{deg}f=4
and is not an even polynomial, then there is a rational point on
. Next, we consider a surface , where
g\in\Q[t] is a monic polynomial of degree six. We prove that if the
polynomial is not even, there is a rational base change
such that on the surface there is a non-torsion section.
Furthermore, if there exists t_{0}\in\Q such that on the curve
there are infinitely many rational points, then
the set of these is infinite. We also present some results concerning
diophantine equation of the form , where is a variable.Comment: 16 pages. Submitted for publicatio
A note on Diophantine systems involving three symmetric polynomials
Let and be -th elementary symmetric polynomial. In this
note we prove that there are infinitely many triples of integers such
that for each the system of Diophantine equations
\begin{equation*}
\sigma_{i}(\bar{X}_{2n})=a, \quad \sigma_{2n-i}(\bar{X}_{2n})=b, \quad
\sigma_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational
solutions. This result extend the recent results of Zhang and Cai, and the
author. Moreover, we also consider some Diophantine systems involving sums of
powers. In particular, we prove that for each there are at least
-tuples of integers with the same sum of -th powers for .
Similar result is proved for and .Comment: to appear in J. Number Theor
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