Questions tagged [abc-conjecture]

The $abc$ conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, $a$, $b$, and $c$ (hence the name) that are relatively prime and satisfy $a + b = c$. If $d$ denotes the product of the distinct prime factors of $abc$, the conjecture essentially states that $d$ is usually not much smaller than $c$. In other words: if $a$ and $b$ are composed of large powers of primes, then $c$ is usually not divisible by large powers of primes. A number of famous conjectures and theorems in number theory would follow immediately from the $abc$ conjecture or its versions. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

The $abc$ conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the $abc$ conjecture. The $abc$ conjecture was shown to be equivalent to the modified Szpiro's conjecture.

The radical of a positive integer $n$, denoted $\mathrm{rad}(n)$, is the product of the distinct prime factors of $n$; for example: $$\mathrm{rad}(27)=\mathrm{rad}(3^3)=3$$ $$\mathrm{rad}(30000)=\mathrm{rad}(2^4\cdot 3^1\cdot 5^4)=2\cdot 3\cdot 5=30$$ $$\mathrm{rad}(2431)=\mathrm{rad}(11^1\cdot 13^1\cdot 17^1)=11\cdot 13\cdot 17=2431$$

Let $a$, $b$ and $c$ be three coprime positive integers such that $a+b=c$. The abc conjecture states that these three equivalent statements are true:

• For every $\varepsilon > 0$, there exist only finitely many triples $(a,b,c)$ such that $$c>\mathrm{rad}(abc)^{1+\varepsilon}$$

• For every $\varepsilon > 0$, there exists a constant $K_{\varepsilon}$ such that $$c < K_{\varepsilon}\cdot\mathrm{rad}(abc)^{1+\varepsilon}$$

• For every $\varepsilon > 0$, there exist only finitely many triples $(a,b,c)$ such that \begin{equation} \frac{\log c}{\log \mathrm{rad}(abc)}>1+\varepsilon \end{equation}

The left side of this inequality is called the quality of a triple $(a,b,c)$ and is denoted $q(a,b,c)$.