Siegel's Theorem

Siegel's Theorem states that for certain types of equations called Diophantine equations, specifically those defining elliptic curves with rational coefficients, there are only finitely many solutions where both variables are rational numbers. In simpler terms, if you look at specific algebraic curves, only a limited number of points on these curves have coordinates that are rational numbers. This result is significant because it shows that, despite the infinite possibilities for solutions in theory, the rational solutions are always finite for these kinds of equations, highlighting a fundamental constraint in number theory.