  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.
