  Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1)
there is an algorithm that for every computable function f:N-->N returns a
positive integer m(f), for which a second algorithm accepts on the input f and
any integer n>=m(f), and returns a system S \subseteq E_n such that S is
consistent over the integers and each integer tuple (x_1,...,x_n) that solves S
satisfies x_1=f(n), (2) there is an algorithm that for every computable
function f:N-->N returns a positive integer w(f), for which a second algorithm
accepts on the input f and any integer n>=w(f), and returns a system S
\subseteq E_n such that S is consistent over N and each tuple (x_1,...,x_n) of
non-negative integers that solves S satisfies x_1=f(n).
