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

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

2011-02-21

arxiv.org/abs/1102.4122v32

Mathematics

Logic

9 pages, the Observation added

Scientific paper

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 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 has infinitely many integer solutions and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n). For each integer n>=12 we construct a system S \subseteq E_n such that S has infinitely many integer solutions and they all belong to Z^n\[-2^{2^{n-1}},2^{2^{n-1}}]^n.

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