ABC conjecture

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

[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 ISBN 0-521-80799-9

[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

[3] Stewart, C. L.; Yu Kunrui; On the abc conjecture. II, Duke Math. J., 108 no. 1 (2001), pp. 169-181, Zbl 1036.11032 ISSN 0012-7094