42,419 research outputs found
On formal inverse of the Prouhet-Thue-Morse sequence
Let be a prime number and consider a -automatic sequence and its generating function
. Moreover, let us
suppose that and and consider the formal power series
which is a compositional inverse of , i.e.,
. In this note we initiate the study of arithmetic
properties of the sequence of coefficients of the power series . We are
mainly interested in the case when , where
and is the
Prouhet-Thue-Morse sequence defined on the two letter alphabet . More
precisely, we study the sequence which is the
sequence of coefficients of the compositional inverse of the generating
function of the sequence . This sequence is clearly 2-automatic. We
describe the sequence characterizing solutions of the equation
. In particular, we prove that the sequence is 2-regular. We
also prove that an increasing sequence characterizing solutions of the equation
is not -regular for any . Moreover, we present a result
concerning some density properties of a sequence related to .Comment: 16 pages; revised version will appear in Discrete Mathematic
On primitive integer solutions of the Diophantine equation and related results
In this paper we investigate Diophantine equations of the form
, where or
and is specific homogenous quintic form. First, we prove that if
and , then the Diophantine equation has solution in
polynomials with integer coefficients, without polynomial common
factor of positive degree. In case we prove that there are
infinitely many primitive integer solutions of the Diophantine equation under
consideration. As an application of our result we prove that for each
n\in\Q\setminus\{0\} the Diophantine equation \begin{equation*}
T^2=n(X_{1}^5+X_{2}^5+X_{3}^5+X_{4}^5) \end{equation*} has a solution in
co-prime (non-homogenous) polynomials in two variables with integer
coefficients. We also present a method which sometimes allow us to prove the
existence of primitive integers solutions of more general quintic Diophantine
equation of the form , where . In particular, we prove that for each the
Diophantine equation \begin{equation*}
T^2=m(X_{1}^5-X_{2}^5)+n^2(X_{3}^5-X_{4}^5) \end{equation*} has a solution in
polynomials which are co-prime over . Moreover, we show how modification
of the presented method can be used in order to prove that for each
n\in\Q\setminus\{0\}, the Diophantine equation \begin{equation*}
t^2=n(X_{1}^5+X_{2}^5-2X_{3}^5) \end{equation*} has a solution in polynomials
which are co-prime over .Comment: 17 pages, submitte
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