### Abstract

Let B n = {x i · x j = x k : i, j, k ∈ {1, . . . , n}} ∪ {x i + 1 = x k : i, k ∈ {1, . . . , n}} denote the system of equations in the variables x 1 , . . . , x n . For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S ⊆ 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 2g(n) for every positive integer n. We conjecture that ξ (n) 6 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 6 d; (4) the conjecture implies that if a set M ⊆ N has a single-fold Diophantine representation, then M is computable; (5) for every integer n > 9, the inequality ξ (n) < (2 2n−5 − 1) 2n−5 + 1 implies that 2 2n−5 + 1 is composite.