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We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdotx_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integersx_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 onlyfinitely many solutions in integers (non-negative integers, rationals), thentheir heights are bounded from above by a computable function of the degree andthe coefficients of the equation. The conjecture implies that the set ofDiophantine equations which have infinitely many solutions in integers(non-negative integers) is recursively enumerable. The conjecture stated for anarbitrary computable bound instead of 2^{2^{n-1}} remains in contradiction toMatiyasevich's conjecture that each recursively enumerable set M \subseteq{\mathbb N}^n has a finite-fold Diophantine representation.

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