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

الأصل · EN

Let Eₙ=xᵢ=1, xᵢ+xⱼ=xₖ, xᵢ · xⱼ=xₖ: i,j,k ∈ 1,...,n. 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 Eₙ such that S has infinitely many integer solutions and each integer tuple (x₁,...,xₙ) that solves S satisfies x₁=f(n). For each integer n>=12 we construct a system S Eₙ such that S has infinitely many integer solutions and they all belong to Zⁿ -2²ⁿ⁻¹,2²ⁿ⁻¹]ⁿ.