Download PDFOpen PDF in browserDefinitive Proof of The abc ConjectureEasyChair Preprint no. 2169, version 211 pages•Date: January 8, 2020AbstractIn this paper, we consider the $abc$ conjecture. Firstly, we give an elementary proof that $c<3rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=\frac{3}{e}.e^{ \left(\frac{1}{\epsilon^2} \right)}$ for $0<\epsilon<1$ and $K(\epsilon)=3$ for $\epsilon \geq 1$. Some numerical examples are presented. Keyphrases: elementary number theory, prime numbers, Real functions of one variable
