Abstract

The height of a rational number p/q is defined by max(|p|,|q|) provided p/q is written in lowest terms. The height of a rational tuple (x_1,...,x_n) is defined as the maximum of n and the heights of the numbers x_1,...,x_n. Let h:\bigcup_{n=1}^\infty Q^n \to N\{0} denote the height function. We conjecture that \forall x_1,...,x_n \in Q \exists y_1,...,y_n \in Q (n=1 ==> h(y_1,...,y_n)=1) \wedge (n \geq 2 ==> h(y_1,...,y_n) \leq 2^(2^(n-2))) \wedge \forall i,j,k \in {1,...,n} ((x_i+1=x_k ==> y_i+1=y_k) \wedge (x_i \cdot x_j=x_k ==> y_i \cdot y_j=y_k)). We prove that the conjecture implies that there is an algorithm which decides whether or not a Diophantine equation has a rational solution.