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 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.

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)

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