ABC conjecture

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

$$\max (|A|,|B|,|C|)\le \kappa (ϵ){(\mathrm{rad}(ABC))}^{1+ϵ}$$

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

The ABC conjecture is consideredone 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.

Further Reading

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)