A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

摘要

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. Weconjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in{1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutionsin non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n)satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture implies thatthere exists an algorithm which takes as input a Diophantine equation, returnsan integer, and this integer is greater than the heights of integer(non-negative integer, positive integer, rational) solutions, if the solutionset is finite, (2) the conjecture implies that the question whether or not aDiophantine equation has only finitely many rational solutions is decidablewith an oracle for deciding whether or not a Diophantine equation has arational solution, (3) the conjecture implies that the question whether or nota Diophantine equation has only finitely many integer solutions is decidablewith an oracle for deciding whether or not a Diophantine equation has aninteger solution, (4) the conjecture implies that if a set M \subseteq N has afinite-fold Diophantine representation, then M is computable.