Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation

•We find functions f1f1, f2f2, g1g1, g2g2 without any finite-fold Diophantine representation.•We state a conjecture on the arithmetic of positive integers.•The conjecture implies that the functions f1f1, f2f2, g1g1, g2g2 are computable.•The conjecture fully solves Diophantine equations with a finite number of solutions.

We define functions f1,f2,g1,g2:N∖{0}→N∖{0}f1,f2,g1,g2:N∖{0}→N∖{0} and prove that they do not have any finite-fold Diophantine representation. We conjecture that if a system S⊆{xi+xj=xk,xi⋅xj=xk:i,j,k∈{1,…,n}} has only finitely many solutions in positive integers x1,…,xnx1,…,xn, then each such solution (x1,…,xn)(x1,…,xn) satisfies x1,…,xn⩽22n−1x1,…,xn⩽22n−1. Assuming the conjecture, we prove: (1) the functions f1f1, f2f2, g1g1, g2g2 are computable, (2) there is 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, (3) a finite-fold Diophantine representation of the function N∋n→2n∈NN∋n→2n∈N does not exist, (4) if a set M⊆NM⊆N is recursively enumerable but not recursive, then a finite-fold Diophantine representation of MM does not exist.