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# Difference between revisions of "ABC conjecture"

m (→Statement: better) |
(→Statement: added Baker's version and Stewart & Yu result) |
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:<math> r(n) = \prod_{p|n} p \ . </math> | :<math> r(n) = \prod_{p|n} p \ . </math> | ||

− | Suppose now that the equation <math>A + B | + | Suppose now that the equation <math>A + B + C = 0</math> holds for 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> | :<math> |A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . </math> | ||

+ | |||

+ | The '''weak ABC conjecture states that | ||

+ | |||

+ | :<math> (|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . </math> | ||

+ | |||

+ | If we define | ||

+ | |||

+ | :<math> \kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \ , | ||

+ | |||

+ | then it is known that <math>\kappa \rightarrow \infty</math> as <math>\espilon \rightarrow 0</math>. | ||

+ | |||

+ | Baker introduced a more refined version of the conjecture in 1996. Assume as before that <math>A + B + C = 0</math> holds for coprime integers <math>A,B,C</math>. Let <math>N</math> be the radical of <math>ABC</math> and <math>\omega</math> the number of distinct prime factors of <math>ABC</math>. Then | ||

+ | |||

+ | :<math> |A|, |B|, |C| < \kappa(\epsilon) (\epsilon^{-\omega} N)^{1+\epsilon} \ . </math> | ||

+ | |||

+ | This form of the conjecture would give very strong bounds in the [[method of linear forms in logarithms]]. | ||

+ | |||

+ | ==Results== | ||

+ | It is known that there is an effectively computable <math>\kappa(\epsilon)</math> such that | ||

+ | |||

+ | :<math> |A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ . </math> |

## Revision as of 18:09, 13 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 an integer to be the product of its distinct prime factors

Suppose now that the equation holds for coprime integers . The conjecture asserts that for every there exists such that

The **weak ABC conjecture states that**

If we define

- as
**Failed to parse (unknown function "\espilon"):**

.

Baker introduced a more refined version of the conjecture in 1996. Assume as before that holds for coprime integers . Let be the radical of and the number of distinct prime factors of . Then

This form of the conjecture would give very strong bounds in the method of linear forms in logarithms.

## Results

It is known that there is an effectively computable such that