Also, I didn't notice this before, but when you say:
> And that seems to be roughly what the abc conjecture says: it is not likely that all of a, b, and c are rare.
It almost sounds like a tautology. Is your view that it is somewhat vacuous after all? Or maybe the interest comes from the fact that while it seems obviously true, we need a proof in order to use it as a theorem for other things where it would be a useful tool...
My intention was to show why you might think it's likely to be true. Maybe I did too good of a job? Heh :)
The justification I gave depends on an intuition that addition "scrambles" prime factorizations, ie. except for common factors, which obviously pass through to the sum, the prime factorization looks random. Understanding how prime factorizations behave under addition is certainly not an easy problem. And then, even if it is unlikely that a, b, and c are all rare, perhaps it is not unlikely in precisely the quantitative way that the abc conjecture supposes.
On the topic of things where it is a useful tool: I found http://www.ams.org/notices/200210/fea-granville.pdf which discusses its relation to other famous results but which also has a much better argument for why it should be true. They approach it as the integer analogue of this result for polynomials
> If a(t), b(t), c(t) ∈ C[t] do not have any common roots and provide a genuine polynomial solution to a(t) + b(t) = c(t), then the maximum of the degrees of a(t), b(t), c(t) is less than the number of distinct roots of a(t) b(t) c(t) = 0.
> And that seems to be roughly what the abc conjecture says: it is not likely that all of a, b, and c are rare.
It almost sounds like a tautology. Is your view that it is somewhat vacuous after all? Or maybe the interest comes from the fact that while it seems obviously true, we need a proof in order to use it as a theorem for other things where it would be a useful tool...