5,302 research outputs found
Perfect powers in products of terms of elliptic divisibility sequences
Diophantine problems involving recurrence sequences have a long history and
is an actively studied topic within number theory. In this paper, we connect to
the field by considering the equation \begin{align*} B_mB_{m+d}\dots
B_{m+(k-1)d}=y^\ell \end{align*} in positive integers with
and , where is a fixed integer and
is an elliptic divisibility sequence, an important class
of non-linear recurrences. We prove that the above equation admits only
finitely many solutions. In fact, we present an algorithm to find all possible
solutions, provided that the set of -th powers in is given. (Note
that this set is known to be finite.) We illustrate our method by an example.Comment: To appear in Bulletin of Australian Math Societ
SOME CONNECTIONS BETWEEN THE SMARANDACHE FUNCTION AND THE FIBONACCI SEQUENCE
This paper is aimed to provide generalizations of the Smarandache function. They
will be constructed by means of sequences more general than the sequence of the
factorials. Such sequences are monotonously convergent to zero sequences and divisibility sequences (in particular the Fibonacci sequence)
Linear divisibility sequences and Salem numbers
We study linear divisibility sequences of order 4, providing a
characterization by means of their characteristic polynomials and finding their
factorization as a product of linear divisibility sequences of order 2.
Moreover, we show a new interesting connection between linear divisibility
sequences and Salem numbers. Specifically, we generate linear divisibility
sequences of order 4 by means of Salem numbers modulo 1
Elliptic divisibility sequences and undecidable problems about rational points
Julia Robinson has given a first-order definition of the rational integers Z
in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0)
where the \forall-quantifiers run over a total of 8 variables, and where F is a
polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove
that a conjecture about elliptic curves provides an interpretation of Z in Q
with quantifier complexity \forall \exists, involving only one universally
quantified variable. This improves the complexity of defining Z in Q in two
ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is
undecidable (recall that Hilbert's Tenth Problem for Q is the question whether
the \Sigma_1-theory of Q is undecidable).
In short, granting the conjecture, there is a one-parameter family of
hypersurfaces over Q for which one cannot decide whether or not they all have a
rational point.
The conjecture is related to properties of elliptic divisibility sequences on
an elliptic curve and its image under rational 2-descent, namely existence of
primitive divisors in suitable residue classes, and we discuss how to prove
weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of
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