Faltings' theorem

Faltings' theorem, also known as the Mordell Conjecture, states that for any algebraic curve (a geometric shape defined by polynomials) with a certain property called genus greater than 1, there are only finitely many rational solutions — meaning only a limited number of points with rational number coordinates. In simpler terms, certain complex equations defining curves have only a limited set of solutions where the coordinates are rational numbers. This groundbreaking result confirms that, in these cases, the solutions are not infinite, providing a deep understanding of the structure of solutions to polynomial equations.