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

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 titl

Open Diophantine Problems

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartie

Why everyone should know number theory

This was an expository lecture for the graduate student colloquium at the University of Arizona on the topic of numbers.Comment: Not for separate publicatio