Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?

Abstract

Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let ξ(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this solution belongs to [1,b]^n. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for every positive integer n. We conjecture that ξ(n) \leq g(2n) for every positive integer n. We prove: (1) the function ξ: N\{0}-->N\{0} is computable in the limit; (2) if a function f:N\{0}-->N\{0} has a single-fold Diophantine representation, then there exists a positive integer m such that f(n)<ξ(n) for every integer n>m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and returns a positive integer d with the following property: for every positive integers a_1,...,a_p, if the tuple (a_1,...,a_p) solely solves the equation D(x_1,...,x_p)=0 in positive integers, then a_1,...,a_p \leq d; (4) the conjecture implies that if a set M \subseteq N has a single-fold Diophantine representation, then M is computable; (5) for every integer n>9, the inequality ξ(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.