Small systems of Diophantine equations which have only very large\n integer solutions

Let E_n={x_i=1, x_i+x_j=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}}. There\nis an algorithm that for every computable function f:N->N returns a positive\ninteger m(f), for which a second algorithm accepts on the input f and any\ninteger n>=m(f), and returns a system S \\subseteq E_n such that S has\ninfinitely many integer solutions and each integer tuple (x_1,...,x_n) that\nsolves S satisfies x_1=f(n). For each integer n>=12 we construct a system S\n\\subseteq E_n such that S has infinitely many integer solutions and they all\nbelong to Z^n\\[-2^{2^{n-1}},2^{2^{n-1}}]^n.\n