# Diophantine Equations and Computation

A Tutorial

Conference paper

## Abstract

Unless otherwise stated, we’ll work with the Consider a Diophantine equation Inverting this, we think of the equation

*natural numbers*:$$N = \{0,1,2,3, \dots\}.$$

*F*(*a*_{1},*a*_{2},...,*a*_{ n },*x*_{1},*x*_{2},...,*x*_{ m }) = 0 with*parameters**a*_{1},*a*_{2},...,*a*_{ n }and*unknowns**x*_{1},*x*_{2},...,*x*_{ m }For such a given equation, it is usual to ask:*For which values of the parameters does the equation have a solution in the unknowns?*In other words, find the set:$$ \{<a_1,\ldots,a_n> \mid \exists x_1,\ldots,x_m [F(a_1,\ldots,x_1,\ldots)=0] \}$$

*F*= 0 furnishing a*definition*of this set, and we distinguish three classes:-
a set is called

*Diophantine*if it has such a definition in which*F*is a polynomial with integer coefficients. We write \(\cal D\) for the*class of Diophantine sets*. -
a set is called

*exponential Diophantine*if it has such a definition in which*F*is an exponential polynomial with integer coefficients. We write \(\cal E\) for the*class of exponential Diophantine sets*. -
a set is called

*recursively enumerable*(or*listable*)if it has such a definition in which*F*is a computable function. We write \(\cal R\) for the*class of recursively enumerable sets*.

## Copyright information

© Springer-Verlag Berlin Heidelberg 2009