The Undecidability of Exponential Diophantine Equations

Abstract

This chapter focuses on the undecidability of exponential Diophantine equations. It is not known whether exponential Diophantine sets are necessarily Diophantine. However, it is known that every exponential Diophantine equation could be transformed mechanically into an equivalent ordinary Diophantine equation in more unknowns, provided that there is a Diophantine equation D (x, y, u 1 , …, u m ) and all solutions of D satisfy y < x x and for every n, there is some solution with y > x n . If such a Diophantine equation exists, then every recursively enumerable set could be Diophantine and Hilbert's tenth problem could be unsolvable. On the other hand, if some particular recursively enumerable set is not Diophantine, it may obtain surprising bounds on the solutions of Diophantine equations. Hilbert''s tenth problem asks for an algorithm to determine whether or not an arbitrary Diophantine equation has a solution in integers. The corresponding problem for exponential Diophantine equations (i.e., equations in which exponentiation as well as addition and multiplication is permitted) can be answered negatively.