On Hilbert's Tenth Problem

Using an iterated Horner schema for evaluation of diophantine polynomials, we define a partial $\mu$-recursive "decision" algorithm decis as a "race" for a first nullstelle versus a first (internal) proof of non-nullity for such a polynomial -- within a given theory T extending Peano Arithmetique PA. If T is diophantine sound, i.e., if (internal) provability implies truth -- for diophantine formulae --, then the T-map decis gives correct results when applied to the codes of polynomial inequalities $D(x_1,...,x_m) \neq 0$. The additional hypothesis that T be diophantine complete (in the syntactical sense) would guarantee in addition termination of decis on these formula, i.e., decis would constitute a decision algorithm for diophantine formulae in the sense of Hilbert's 10th problem. From Matiyasevich's impossibility for such a decision it follows, that a consistent theory T extending PA cannot be both diophantine sound and diophantine complete. We infer from this the existence of a diophantine formulae which is undecidable by T. Diophantine correctness is inherited by the diophantine completion T~ of T, and within this extension decis terminates on all externally given diophantine polynomials, correctly. Matiyasevich's theorem -- for the strengthening T~ of T -- then shows that T~, and hence T, cannot be diophantine sound. But since the internal consistency formula Con_T for T implies -- within PA -- diophantine soundness of T, we get that PA derives \neg Con_T, in particular PA must derive its own internal inconsistency formula

On products of disjoint blocks of arithmetic progressions and related equations

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erd\H{o}s and Graham.Comment: 10 page

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

On the Diophantine Equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the exponential Diophantine equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h into subcases that are easier to analyze. Then we will solve an equation obtained by imposing a restriction on one of these subcases, after which we will solve a generalization of this equation.Comment: This 6-page paper is the second part of an honors thesis I have written as an undergraduate at UC Berkele

On the cohomological equation for nilflows

Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions D_n such that any sufficiently smooth function g is a coboundary iff it belongs to the kernel of all the distributions D_n.Comment: 27 page