101 research outputs found

    On certain arithmetic properties of Stern polynomials

    Full text link
    We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: B0(t)=0,B1(t)=1,B2n(t)=tBn(t)B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t), and B2n+1(t)=Bn(t)+Bn+1(t)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 del Pezzo surfaces of degree one

    Full text link
    Let f(z)=z5+az3+bz2+cz+dZ[z]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 Ef:x2y3f(z)=0\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 Ea,b:Y2=X3+135(2a15)X1350(5a+2b26)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 Ef\cal{E}_{f} is dense in the Zariski topology.Comment: 8 pages. Published in Glasgow Mathematical Journa

    Rational points on certain hyperelliptic curves over finite fields

    Full text link
    Let KK be a field, a,bKa, b\in K and ab0ab\neq 0. Let us consider the polynomials g1(x)=xn+ax+b,g2(x)=xn+ax2+bxg_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx, where nn is a fixed positive integer. In this paper we show that for each k2k\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 Ci:y2=gi(x),(i=1,2)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

    On formal inverse of the Prouhet-Thue-Morse sequence

    Full text link
    Let pp be a prime number and consider a pp-automatic sequence u=(un)nN{\bf u}=(u_{n})_{n\in\N} and its generating function U(X)=n=0unXnFp[[X]]U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]. Moreover, let us suppose that u0=0u_{0}=0 and u10u_{1}\neq 0 and consider the formal power series VFp[[X]]V\in\mathbb{F}_{p}[[X]] which is a compositional inverse of U(X)U(X), i.e., U(V(X))=V(U(X))=XU(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)V(X). We are mainly interested in the case when un=tnu_{n}=t_{n}, where tn=s2(n)(mod2)t_{n}=s_{2}(n)\pmod{2} and t=(tn)nN{\bf t}=(t_{n})_{n\in\N} is the Prouhet-Thue-Morse sequence defined on the two letter alphabet {0,1}\{0,1\}. More precisely, we study the sequence c=(cn)nN{\bf c}=(c_{n})_{n\in\N} which is the sequence of coefficients of the compositional inverse of the generating function of the sequence t{\bf t}. This sequence is clearly 2-automatic. We describe the sequence a{\bf a} characterizing solutions of the equation cn=1c_{n}=1. In particular, we prove that the sequence a{\bf a} is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation cn=0c_{n}=0 is not kk-regular for any kk. Moreover, we present a result concerning some density properties of a sequence related to a{\bf a}.Comment: 16 pages; revised version will appear in Discrete Mathematic

    Rational solutions of certain Diophantine equations involving norms

    Full text link
    In this note we present some results concerning the unirationality of the algebraic variety Sf\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 kk is a number field, K=k(α)K=k(\alpha), α\alpha is a root of an irreducible polynomial h(x)=x3+ax+bk[x]h(x)=x^3+ax+b\in k[x] and fk[t]f\in k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a=0a=0 and bkk3b\in k\setminus k^{3}. We prove that if \op{deg}f=4 and the variety Sf\cal{S}_{f} contains a kk-rational point (x0,y0,z0,t0)(x_{0},y_{0},z_{0},t_{0}) with f(t0)0f(t_{0})\neq 0, then Sf\cal{S}_{f} is kk-unirational. A similar result is proved for a broad family of quintic polynomials ff satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of Sf\cal{S}_{f} (with non-trivial kk-rational point) is proved for any polynomial ff of degree 6 with ff not equivalent to the polynomial hh satisfying the condition h(t)h(ζ3t)h(t)\neq h(\zeta_{3}t), where ζ3\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)=x3+ax+bk[x]h(x)=x^3+ax+b\in k[x], provided that f(t)=t6+a4t4+a1t+a0k[t]f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t] with a1a40a_{1}a_{4}\neq 0.Comment: submitte

    Some experiments with Ramanujan-Nagell type Diophantine equations

    Get PDF
    Stiller proved that the Diophantine equation x2+119=152nx^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 x2=Akn+Bx^2=Ak^{n}+B with many solutions. Here, A,BZA,B\in\Z (thus A,BA, B are not necessarily positive) and kZ2k\in\Z_{\geq 2} are given integers. In particular, we prove that for each kk there exists an infinite set S\cal{S} containing pairs of integers (A,B)(A, B) such that for each (A,B)S(A,B)\in \cal{S} we have gcd(A,B)\gcd(A,B) is square-free and the Diophantine equation x2=Akn+Bx^2=Ak^n+B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x2=Akn+Bx^2=Ak^n+B with k>2k>2, each containing five solutions in non-negative integers. %For example the equation y2=1303n+5550606y^2=130\cdot 3^{n}+5550606 has exactly five solutions with n=0,6,11,15,16n=0, 6, 11, 15, 16. We also find new examples of equations x2=A2n+Bx^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

    Full text link
    Let Ef:y2=x3+f(t)x\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 tϕ(t)t\mapsto\phi(t) such that on the surface Efϕ\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 Et0:y2=x3+f(t0)xE_{t_{0}}:y^2=x^3+f(t_{0})x. In particular, we prove that if \op{deg}f=4 and ff is not an even polynomial, then there is a rational point on Ef\cal{E}_{f}. Next, we consider a surface Eg:y2=x3+g(t)\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 gg is not even, there is a rational base change tψ(t)t\mapsto\psi(t) such that on the surface Egψ\cal{E}^{g\circ\psi} there is a non-torsion section. Furthermore, if there exists t_{0}\in\Q such that on the curve Et0:y2=x3+g(t0)E^{t_{0}}:y^2=x^3+g(t_{0}) there are infinitely many rational points, then the set of these t0t_{0} is infinite. We also present some results concerning diophantine equation of the form x2y3g(z)=tx^2-y^3-g(z)=t, where tt 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

    Full text link
    In this paper we investigate Diophantine equations of the form T2=G(X),  X=(X1,,Xm)T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m}), where m=3m=3 or m=4m=4 and GG is specific homogenous quintic form. First, we prove that if F(x,y,z)=x2+y2+az2+bxy+cyz+dxzZ[x,y,z]F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,y,z] and (b2,4ad2,d)(0,0,0)(b-2,4a-d^2,d)\neq (0,0,0), then the Diophantine equation t2=nxyzF(x,y,z)t^2=nxyzF(x,y,z) has solution in polynomials x,y,z,tx, y, z, t with integer coefficients, without polynomial common factor of positive degree. In case a=d=0,b=2a=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 T2=aX15+bX25+cX35+dX45T^2=aX_{1}^5+bX_{2}^5+cX_{3}^5+dX_{4}^5, where a,b,c,dZa, b, c, d\in\Z. In particular, we prove that for each m,nZ{0},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]\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]\Z[t].Comment: 17 pages, submitte
    corecore