Unsolvable diophantine problems

Abstract

We shall show that there is no general method of telling whether an arbitrary polynomial P(xl, , Xk) with integer coefficients is ever a power of 2 for xi, , xk natural numbers. At present there is no general method known even in the special case with k = 1. Actually we shall show directly that the relation given by r =2 is diophantine in the set S of powers of 2. Hence every recursively enumerable set is diophantine in i by Corollary 5 of Davis, Putnam, and Robinson [4]. We call a relation p(x1, , xn) among natural numbers diophantine if there is a polynomial P with integer coefficients such that p(xi, , xn) if and only if there are natural numbers yi, , Yk with P(xi, , xn, yi, , yk) = 0. A set 8 (function F) is diophantine if xCS (the graph of F) is Also, p is diophantine in a set 8 if there is a polynomial P with integer coefficients such that p(xi, , xn) if and only if there are natural numbers yi, , Yk, Z, ... z1 with both P(xi, , xn, Yl, , Yk, zl, , z,)=0 and Zi, , z,CS. In [6], the term existentially definable was used instead of diophantine. The definitions given there are easily seen to be equivalent to these. In both [1] and [4], relations over the positive integers were considered but the definitions and theorems hold for natural numbers with only the obvious modifications. The logical symbols V (there exists), A (and), 1 and an, a ' be defined by