Whether one of these entities has a finite number of rational solutions can be determined by the topology of an algebraic surface created by allowing variables to assume complex values, in Falting's theorem.
Whether one of these entities has a finite number of rational solutions can be determined by the topology of an algebraic surface created by allowing variables to assume complex values, in Falting's theorem. Baker's method sets the upper bound for their solutions, but only holds for the case of two variables. A set of simultaneous ones can be used to solve Archimedes' Cattle problem. Hilbert's tenth problem was aimed at the development of an algorithm to determine whether an arbitrary one has a solution and Catalan's conjecture proposes a solution to a specific case. Gauss provided a procedure for determining solutions of the quadratic type in two variables through reduction to the Fermat difference equation. For 10 points, the Bachet and Pell equations are special cases of what type of equation in which only integer solutions are allowed?
Whether one of these entities has a finite number of rational solutions can be determined by the topology of an algebraic surface created by allowing variables to assume complex values, in Falting's theorem.
{Diophantine} equations