Questions of decidability and undecidability in Number Theory

Abstract

Davis, Matijasevic, and Robinson, in their admirable survey article [D-M-R], interpret the negative solution of Hilbert's Tenth Problem as a resounding positive statement about the versatility of Diophantine equations (that any listable set can be coded as the set of parameter values for which a suitable polynomial possesses integral solutions).One can also view the Matijasevic result as implying that there are families of Diophantine equations parametrized by a variable t, which have integral solutions for some integral values t = a > 0, and yet there is no computable function of t which provides an upper bound for the smallest integral solution for these values a. The smallest integral solutions of the Diophantine equation for these values are, at least sporadically, too large to be bounded by any computable function. This is somewhat difficult to visualize, since there is quite an array of computable functions. But let us take an explicit example. Consider the functionMatijasevic's result guarantees the existence of parametrized families of Diophantine equations such that even this function fails to yield an upper bound for its smallest integral solutions (for all values of the parameter t for which there are integral solutions).Families of Diophantine equations in a parameter t, whose integral solutions for t = 1, 2, 3,… exhibit a certain arythmia in terms of their size, have fascinated mathematicians for centuries, and this phenomenon (the size of smallest integral solution varying wildly with the parameter-value) is surprising, even when the equations are perfectly “decidable”.