The abc conjecture is essentially an analogue of the Mason-Stothers Theorem for integers. Roughly, the idea is that if two integers are products of large powers of small primes, then their sum should be a product of small powers of large primes. To translate the theorem more precisely, we replace the number of distinct roots of a polynomial by the radical of an integer, that is, the product of its prime factors, $\mathrm{rad}\left(n\right)equals;\prod _{pverbar;n}^{}p$ ; and we replace the degree of a polynomial by the absolute value of an integer. So the precise analogue would be:
For any triple of coprime positive integers $a\mathit{\,}b\mathit{\,}c$ with $a\mathit{\+}b\mathit{}\mathit{equals;}c$, $\mathrm{rad}\left(a\mathit{}b\mathit{}c\right)\mathit{}\mathit{gt;}\mathit{}c$. (This is not actually true!)
In fact there are known to be infinitely many examples of such triples with $\mathrm{rad}\left(abc\right)c$. However, the actual conjecture states that the previous statement is almost true, in the following sense. If we just raise the threshold to the power $1\+\mathrm{\ε}$, no matter how small we make $\mathrm{\ε}$, then there are only finitely many counterexamples:
abc Conjecture. For every $\mathrm{\ε}\mathit{}\mathit{gt;}\mathit{}\mathit{0}$, there are only finitely many triples of coprime positive integers $a\mathit{\,}b\mathit{\,}c$ with $a\mathit{\+}b\mathit{}\mathit{equals;}c$ and $\mathrm{rad}{\left(a\mathit{}b\mathit{}c\right)}^{\mathit{1}\mathit{plus;}\mathrm{epsilon;}}\mathit{}\mathit{}\mathit{}\mathit{}c$.
The abc conjecture was proposed by Oesterlé and Masser in 1985, and it has since been determined to be both extremely difficult to prove, and at the same time very important! Indeed, it has been shown that if the abc Conjecture is true, then a number of important results in number theory, including Fermat's Last Theorem, would follow as consequences.