Reductions of Hilbert's Tenth Problem

(1) (Eyl)... (Eym)[P(yi, Y2, ** Ymin X1, X2, xn) = 01 where P is a polynomial with integral coefficients (positive, negative, or zero). Previous work' has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recfirsive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less. For example, we have: THEOREM 1. If some recursively enumerable and non-recursive predicate H(x1, ..., xn) is expressible in the form