ABC conjecture: Difference between revisions
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In mathematics, the '''ABC conjecture''' relates the prime factors of two integers to those of their sum. It was proposed by [[David Masser]] and [[Joseph Oesterlé]] in 1985. It is connected with other problems of [[number theory]]: for example, the truth of the ABC conjecture would provide a new proof of [[Fermat's Last Theorem]]. | |||
==Statement== | |||
Define the ''radical'' of a number to be the product of its distinct prime factors | |||
:<math> r(n) = \prod_{p|n} p \ . </math> | |||
Suppose now that the equation <math>A + B = C</math> holds for positive coprime integers <math>A,B,C</math>. The conjecture asserts that for every <math>\epsilon > 0</math> there exists <math>\kappa(\epsilon) > 0</math> such that | |||
:<math> |A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . </math> | |||
Revision as of 15:30, 11 January 2013
In mathematics, the ABC conjecture relates the prime factors of two integers to those of their sum. It was proposed by David Masser and Joseph Oesterlé in 1985. It is connected with other problems of number theory: for example, the truth of the ABC conjecture would provide a new proof of Fermat's Last Theorem.
Statement
Define the radical of a number to be the product of its distinct prime factors
Suppose now that the equation holds for positive coprime integers . The conjecture asserts that for every there exists such that
