101 research outputs found

    On certain arithmetic properties of Stern polynomials

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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