# ABC conjecture

The ABC conjecture states that given any $\epsilon>0$, there is a constant $\kappa(\epsilon)$ such that

 $\max(|A|,|B|,|C|)\leq\kappa(\epsilon)(\operatorname{rad}(ABC))^{1+\epsilon}$

for all mutually coprime integers $A$, $B$, $C$ with $A+B=C$, where $\operatorname{rad}$ is the radical of an integer. This conjecture was formulated by Masser and Oesterlé in 1980.

The ABC conjecture is considered one of the most important unsolved problems in number , 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.

http://www.maa.org/mathland/mathtrek_12_8.htmlThe Amazing ABC Conjecture — an article on the ABC conjecture by Ivars Peterson.

http://www.hcs.harvard.edu/hcmr/issue1/elkies.pdfThe ABC’s of Number Theory — an article on the ABC conjecture by Noam Elkies. (PDF file)

 Title ABC conjecture Canonical name ABCConjecture Date of creation 2013-03-22 11:45:23 Last modified on 2013-03-22 11:45:23 Owner yark (2760) Last modified by yark (2760) Numerical id 21 Author yark (2760) Entry type Conjecture Classification msc 11A99 Classification msc 55-00 Classification msc 82-00 Classification msc 83-00 Classification msc 81-00 Classification msc 18-00 Classification msc 18C10