MuPAD codes which implement limit-computable functions that cannot be bounded by any computable function

Let E_n = {x_k = 1, x_i + x_j = x_k, x_i · x_j = x_k : i, j, k ∈ {1, ..., n}}. For a positive integer n, let f (n) denote the smallest non-negative integer b such that for each system S ⊆ E_n with a solution in non-negative integers x₁, ..., x_n there exists a solution of S in non-negative integers not greater than b. We prove that if a function Γ : ℕ {0} → ℕ is computable, then f dominates Γ i.e. there exists a positive integer m such that Γ(n) <; f (n) for any n ≥ m. For positive integers n, m, let g(n,m) denote the smallest non-negative integer b such that for each system S ⊆ E_n with a solution in {0, ..., m - 1}ⁿ there exists a solution of S in {0, ..., b}ⁿ. Then, equations and equation We present an infinite loop in MuPAD which takes as input a positive integer n and returns g(n,m) on the m-th iteration.