I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you kind people could help me.
I know there are several formulations of the conjecture.
Wolfram says:
for any infinitesimal $\epsilon > 0$, there exists a constant $C_\epsilon$ such that for any three relatively prime integers $a$, $b$, $c$ satisfying $a+b=c$ the inequality $$\max (|a|, |b|, |c|) \leq C_{\epsilon}\displaystyle\prod_{p|abc} p^{1+\epsilon}$$
holds, where $p|abc$ indicates that the product is over primes $p$ which divide the product $abc$.
Then Wikipedia says:
For a positive integer $n$, the radical of $n$, denoted $\text{rad}(n)$, is the product of the distinct prime factors of $n$. If $a$, $b$, and $c$ are coprime positive integers such that $a + b = c$, it turns out that "usually" $c < \text{rad}(abc)$. The abc conjecture deals with the exceptions. Specifically, it states that for every $\epsilon>0$ there exist only finitely many triples $(a,b,c)$ of positive coprime integers with $a + b = c$ such that $$c>\text{rad}(abc)^{1+\epsilon}$$
An equivalent formulation states that for any $\epsilon > 0$, there exists a constant $K$ such that, for all triples of coprime positive integers $(a, b, c)$ satisfying $a + b = c$, the inequality $$c<K\cdot\text{rad}(abc)^{1+\epsilon}$$
holds.
A third formulation of the conjecture involves the quality $q(a, b, c)$ of the triple $(a, b, c)$, defined by: $$q(a,b,c)=\frac{\log(c)}{\log(\text{rad}(abc)}$$
I am particularly interested in the first definition, but any help with any of it would be greatly appreciated.
Best Answer
Try Mazur's Questions about Number (1995).