"Define the function Hud in number theory, of which the abc conjecture is a minor consequence."

In this paper, we investigate the function Hudk(n), which is de f ined using prime factorization with restricted exponents. We provide its definition, prove inequalities related to Hudk, and demonstrate applications to various famous problems in number theory, including Fermat’s Last Theorem, the abc conjecture, Roth’s theorem, Mordell’s conjecture, Catalan’s conjecture, and others. The function Huda rves as a unifying framework to derive bounds and explore deep connections between exponential Diophantine equations and prime factorization structures<br><br>