A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions

Apoloniusz Tyszka

Published 2009-01-14, updated 2014-10-20Version 152

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq 2^{2^{n-1}}. By the conjecture, if a Diophantine equation has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of the degree and the coefficients of the equation. The conjecture implies that the set of Diophantine equations which have infinitely many solutions in integers (non-negative integers) is recursively enumerable. The conjecture stated for an arbitrary computable bound instead of 2^{2^{n-1}} remains in contradiction to Matiyasevich's conjecture that each recursively enumerable set M \subseteq {\mathbb N}^n has a finite-fold Diophantine representation.