# ABC conjecture

The *ABC conjecture ^{}* states that given any $\u03f5>0$,
there is a constant $\kappa (\u03f5)$ such that

$$\mathrm{max}(|A|,|B|,|C|)\le \kappa (\u03f5){(\mathrm{rad}(ABC))}^{1+\u03f5}$$ |

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 |