ABCConjecture 2001-10-15 20:02:54 2007-07-05 02:53:27 Conjecture number theory \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\rad}{\operatorname{rad}} The \emph{ABC conjecture} states that given any $\epsilon > 0$, there is a constant $\kappa ( \epsilon )$ such that \[ \max(|A|,|B|,|C|) \leq \kappa ( \epsilon ) ( \rad (ABC))^{1 + \epsilon} \] for all mutually coprime integers $A$, $B$, $C$ with $A+B=C$, where $\rad$ is the radical of an integer. This conjecture was formulated by Masser and Oesterl\'{e} in 1980. The ABC conjecture is considered one of the most important unsolved problems in number \PMlinkescapetext{theory}, as many results would follow directly from this conjecture. For example, Fermat's Last Theorem could be proved (for sufficiently large exponents) with about one page worth of proof. \section*{Further Reading} \PMlinkexternal{The Amazing ABC Conjecture}{http://www.maa.org/mathland/mathtrek_12_8.html} --- an article on the ABC conjecture by Ivars Peterson. \PMlinkexternal{The ABC's of Number Theory}{http://www.hcs.harvard.edu/hcmr/issue1/elkies.pdf} --- an article on the ABC conjecture by Noam Elkies. (PDF file)