549 research outputs found
Rational Points in Geometric Progression on the Unit Circle
A sequence of rational points on an algebraic planar curve is said to form an
-geometric progression sequence if either the abscissae or the ordinates of
these points form a geometric progression sequence with ratio . In this
work, we prove the existence of infinitely many rational numbers such that
for each there exist infinitely many -geometric progression sequences on
the unit circle of length at least .Comment: 7 pages, accepted for publication in Publicationes Mathematicae
Debrece
On two four term arithmetic progressions with equal product
We investigate when two four-term arithmetic progressions have an equal
product of their terms. This is equivalent to studying the (arithmetic) geometry of a non-singular quartic surface. It turns out that there are many
polynomial parametrizations of such progressions, and it is likely that there
exist polynomial parametrizations of every positive degree. We find all such
parametrizations for degrees 1 to 4, and give examples of parametrizations
for degrees 5 to 10
Circles with four rational points in geometric progression
A set of rational points on a curve is said to be in geometric progression if
either the abscissae or the ordinates of the points are in geometric
progression. Examples of three points in geometric progression on a circle are
already known. In this paper we obtain infinitely many examples of four points
in geometric progression on a circle with rational radius.Comment: 11 page
Polynomial sequences on quadratic curves
In this paper we generalize the study of Matiyasevich on integer points over
conics, introducing the more general concept of radical points. With this
generalization we are able to solve in positive integers some Diophantine
equations, relating these solutions by means of particular linear recurrence
sequences. We point out interesting relationships between these sequences and
known sequences in OEIS. We finally show connections between these sequences
and Chebyshev and Morgan-Voyce polynomials, finding new identities
The Brauer-Manin Obstruction and Sha[2].
We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor
Average Bateman--Horn for Kummer polynomials
For any and almost all smaller than
, we show that the polynomial takes the expected number
of prime values as ranges from 1 to . As a consequence, we deduce
statements concerning variants of the Hasse principle and of the integral Hasse
principle for certain open varieties defined by equations of the form
where is a
quadratic extension. A key ingredient in our proof is a new large sieve
inequality for Dirichlet characters of exact order .Comment: V2: Minor correction
Linear relations with conjugates of a Salem number
In this paper we consider linear relations with conjugates of a Salem number
. We show that every such a relation arises from a linear relation
between conjugates of the corresponding totally real algebraic integer
. It is also shown that the smallest degree of a Salem number
with a nontrivial relation between its conjugates is , whereas the smallest
length of a nontrivial linear relation between the conjugates of a Salem number
is .Comment: v1, 12 page
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