Hilbert's Tenth Problem

Hilbert's Tenth Problem asks whether there's a universal method—an algorithm—that can, for any given equation with whole numbers, determine if the equation has solutions or not. Specifically, it focuses on polynomial equations with integer coefficients and seeks a way to decide solvability systematically. In the 20th century, mathematicians proved that no such general algorithm exists; the problem is undecidable. This means some equations cannot be definitively labeled as solvable or insoluble using a finite procedure, revealing fundamental limits in what computers can solve in number theory.