A new characterization of computable functions

Let E_n ={x_i = 1; x_i + x_j = x_k; x_i ∙ x_j = x_k : i; j; k ∊ {1;. . . ; n}}. We present two algorithms. The first accepts as input any computable function f : ℕ → ℕ and returns a positive integer m( f ) and a computable function g which to each integer n ≥ m( f ) assigns a system S ⊆ E_n such that S is satisfiable over integers and each integer tuple (x₁;. . .; x_n) that solves S satisfies x₁ = f (n). The second accepts as input any computable function f: ℕ → ℕ and returns a positive integer w( f ) and a computable function h which to each integer n ≥ w( f ) assigns a system S ⊆ E_n such that S is satisfiable over non-negative integers and each tuple (x₁;. . .; x_n) of non-negative integers that solves S satisfies x₁ = f (n).