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

Let Bn = {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈ {1, . . . , n}} denote the system of equations in the variables x1, . . . , xn. For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S ⊆ Bn with a unique solution in positive integers x1, . . . , xn, this solution belongs to [1, b]n. Let g(1) = 1, and let g(n + 1) = 22g(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(x1, . . . , xp) = 0 and returns a positive integer d with the following property: for every positive integers a1, . . . , ap, if the tuple (a1, . . . , ap) solely solves the equation D(x1, . . . , xp) = 0 in positive integers, then a1, . . . , ap 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) < (22n−5 − 1)2n−5 + 1 implies that 22n−5 + 1 is composite.

Walking Cautiously Into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly

International audience Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: \emphalgorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their \emphcyclic and divergent properties.