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The height of a rational number p/q is denoted by h(p/q) and equals max (|p|, |q|) provided is written in lowest terms. The height of a rational tuple h(x1,..., xn) is denoted by h(x1,..., xn) and equals max(h(x1),..., h(xn)). Let Gn = {xi + 1 = xk : i, k &#x03F5; {1,..., n}} &#x222A; (xi, &#x00B7; xj = xk : i, j, k &#x03F5; {1,..., n}}. Let f (1) = 1, and let f (n + 1) = 29f(n) for every positive integer n. We conjecture: (1) if a system S &#x2286; Gn has only finitely many solutions in rationals x1,..., xn, then each such solution (x1,..., xn) satisfies h(x1,..., xn) &#x2264; {1 (if n = 1) 22n-2 (if &#x2265; 2) if a system S &#x2286; Gn has only finitely many solutions in non-negative rationals x1,..., xn, then each such solution (x1,..., xn) satisfies h(x1,..., xn) &#x2264; f (2n). We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution.

Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?