  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 \xi(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 \xi(n) \leq g(2n) for every
positive integer n. We prove: (1) the function \xi: 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)<\xi(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
\xi(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.

Logic;Number Theory