View source for ABC conjecture

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A conjectural relationship between the prime factors of two  integers and those of their sum, 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]].  

Define the ''radical'' of an integer to be the product of its distinct prime factors
$$
r(n) = \prod_{p|n} p \ .
$$
Suppose  now that the equation $A + B + C = 0$ holds for  coprime integers $A,B,C$.  The conjecture asserts  that for every $\epsilon > 0$ there exists $\kappa(\epsilon) > 0$ such that
$$
|A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . 
$$
A weaker form of the conjecture states that
$$
(|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . 
$$
If we define 
$$
\kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \,,
$$
then it is known that $\kappa \rightarrow \infty$ as $\epsilon \rightarrow 0$.

Baker  introduced a more refined version of the conjecture in 1998.  Assume as  before that $A + B + C = 0$ holds for coprime  integers $A,B,C$.  Let $N$  be the radical of $ABC$ and $ \omega$ the number of distinct prime factors of $ABC$.  Then there is an absolute constant$c$  such that
$$
|A|, |B|, |C| < c (\epsilon^{-\omega} N)^{1+\epsilon} \ .
$$

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

It is known that there is an effectively computable $\kappa(\epsilon)$ such that 
$$
|A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ . 
$$

==References==
* {{Ref|1}}  Goldfeld, Dorian; "Modular forms, elliptic curves and the abc-conjecture", ed. Wüstholz, Gisbert; ''A panorama in number theory or The view from Baker's garden'', (2002), pp. 128-147, Cambridge University Press {{ZBL|1046.11035}} {{ISBN|0-521-80799-9}}
* {{Ref|2}}  Baker, Alan; "Logarithmic forms and the abc-conjecture", ed. Győry, Kálmán (ed.) et al.; ''Number  theory. Diophantine, computational and algebraic aspects. Proceedings  of the international conference, Eger, Hungary, July 29-August 2, 1996'',     (1998), pp. 37-44, de Gruyter, {{ZBL|0973.11047}} {{ISBN|3-11-015364-5}}
* {{Ref|3}}  Stewart, C. L.; Yu Kunrui; ''On the abc conjecture. II'', Duke Math. J.,  108 no. 1 (2001), pp. 169-181, {{ZBL|1036.11032}} {{DOI|10.1215/S0012-7094-01-10815-6}}

[[Category:Number theory]]