A hypothetical upper bound on the heights of the solutions of a   Diophantine equation with a finite number of solutions
  Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture 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 solutions in 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 that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite, (2) the conjecture implies that the question whether or not a Diophantine equation has only finitely many rational solutions is decidable with an oracle for deciding whether or not a Diophantine equation has a rational solution, (3) the conjecture implies that the question whether or not a Diophantine equation has only finitely many integer solutions is decidable with an oracle for deciding whether or not a Diophantine equation has an integer solution, (4) the conjecture implies that if a set M \subseteq N has a finite-fold Diophantine representation, then M is computable. 
Apoloniusz Tyszka
math.NT