On certain arithmetic properties of Stern polynomials

We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. 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 $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ 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 $C_{i}:y^2=g_{i}(x), (i=1, 2)$ 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 $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a, b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.Comment: 8 pages. Published in Glasgow Mathematical Journa

On formal inverse of the Prouhet-Thue-Morse sequence

Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and consider the formal power series $V\in\mathbb{F}_{p}[[X]]$ which is a compositional inverse of $U(X)$, i.e., $U(V(X))=V(U(X))=X$. In this note we initiate the study of arithmetic properties of the sequence of coefficients of the power series $V(X)$. We are mainly interested in the case when $u_{n}=t_{n}$, where $t_{n}=s_{2}(n)\pmod{2}$ and ${\bf t}=(t_{n})_{n\in\N}$ is the Prouhet-Thue-Morse sequence defined on the two letter alphabet $\{0,1\}$. More precisely, we study the sequence ${\bf c}=(c_{n})_{n\in\N}$ which is the sequence of coefficients of the compositional inverse of the generating function of the sequence ${\bf t}$. This sequence is clearly 2-automatic. We describe the sequence ${\bf a}$ characterizing solutions of the equation $c_{n}=1$. In particular, we prove that the sequence ${\bf a}$ is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation $c_{n}=0$ is not $k$-regular for any $k$. Moreover, we present a result concerning some density properties of a sequence related to ${\bf a}$.Comment: 16 pages; revised version will appear in Discrete Mathematic

Rational solutions of certain Diophantine equations involving norms

In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+\alpha X_{2}+\alpha^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(\alpha)$, $\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if \op{deg}f=4 and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirational. A similar result is proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $\cal{S}_{f}$ (with non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to the polynomial $h$ satisfying the condition $h(t)\neq h(\zeta_{3}t)$, where $\zeta_{3}$ 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 $h(x)=x^3+ax+b\in k[x]$, provided that $f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t]$ with $a_{1}a_{4}\neq 0$.Comment: submitte

Some experiments with Ramanujan-Nagell type Diophantine equations

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five solutions in non-negative integers. %For example the equation $y^2=130\cdot 3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also find new examples of equations $x^2=A2^{n}+B$ 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 $\mathcal{E}_{f}:y^2=x^3+f(t)x$, 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 $t\mapsto\phi(t)$ such that on the surface $\cal{E}_{f\circ\phi}$ 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 $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if \op{deg}f=4 and $f$ is not an even polynomial, then there is a rational point on $\cal{E}_{f}$. Next, we consider a surface $\cal{E}^{g}:y^2=x^3+g(t)$, where g\in\Q[t] is a monic polynomial of degree six. We prove that if the polynomial $g$ is not even, there is a rational base change $t\mapsto\psi(t)$ such that on the surface $\cal{E}^{g\circ\psi}$ there is a non-torsion section. Furthermore, if there exists t_{0}\in\Q such that on the curve $E^{t_{0}}:y^2=x^3+g(t_{0})$ there are infinitely many rational points, then the set of these $t_{0}$ is infinite. We also present some results concerning diophantine equation of the form $x^2-y^3-g(z)=t$, where $t$ is a variable.Comment: 16 pages. Submitted for publicatio

On primitive integer solutions of the Diophantine equation t2=G(x,y,z)t^2=G(x,y,z) and related results

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if $F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,y,z]$ and $(b-2,4a-d^2,d)\neq (0,0,0)$, then the Diophantine equation $t^2=nxyzF(x,y,z)$ has solution in polynomials $x, y, z, t$ with integer coefficients, without polynomial common factor of positive degree. In case $a=d=0, b=2$ 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 $T^2=aX_{1}^5+bX_{2}^5+cX_{3}^5+dX_{4}^5$, where $a, b, c, d\in\Z$. In particular, we prove that for each $m, n\in\Z\setminus\{0\},$ 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 $\Z[t]$. 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 $\Z[t]$.Comment: 17 pages, submitte