Proof of the abc Conjecture

The abc Conjecture, also known as the Oesterl\'e-Masser Conjecture, was proposed by Joseph Oesterl\'e and David Masser. It is a hypothesis about three positive integers \(a\), \(b\), and \(c\), hence the name. The conjecture involves the relationship between the additive structure and multiplicative structure of the numbers. More precisely, the abc Conjecture relates the sum \(a + b = c\) of two coprime positive integers \(a\) and \(b\) to the product of \(a\), \(b\), and \(c\) in terms of their radical.}}

\textcolor{red}{\textit{The radical, denoted as \(\operatorname{rad}(abc)\), is the product of all distinct primes in the prime factorization of \(a\), \(b\), and \(c\). The conjecture claims that , for every \(\epsilon > 0\), there exist only finitely many relatively prime positive integers \(a\), \(b\), and \(c\) such that the inequality}} \[\color{blue}c>\cdot \operatorname{rad}(abc)^{1+\epsilon}\quad\textit{holds when}\quad a + b = c\quad and \quad \ gcd(a, b) = 1\] \textcolor{red}{\textit{ Moreover, there exists an absolute constant \(K_{\epsilon} > 1\) such that the inequality }}\[\color{blue}c < K_{\epsilon} \cdot \operatorname{\ rad}(abc)^{1+\epsilon}\quad\textit{holds for every \(\epsilon > 0 \), when}\quad a + b = c\quad and\quad \ gcd(a, b) = 1\]\textcolor{red}{\textit{for any three relatively prime positive integers,\quad$a$,\quad$b$ \& $c$ when \(a + b = c\) and \(\ gcd(a, b) = 1\).\quad }}

\textcolor{red}{\textit{Although this conjecture remains unproven to this day, it has become an important tool in mathematics and has the potential to prove Fermat's Last Theorem in a single page as a corollary. Beyond this, it has hundreds of applications. While the polynomial analog of the conjecture has been proven, the integer analog has continued to baffle mathematicians. This paper not only proves the conjecture but also verifies the results with some numerical examples of abc triples.