4 ms·

Another way to think about it is to consider what needs to happen for rad(abc) to be much less than c. This can only happen if the "dropping" you mention drops

Another way to think about it is to consider what needs to happen for rad(abc) to be much less than c. This can only happen if the "dropping" you mention drops a whole lot of factors. In other words, it can only happen if a, b, and c all contain mostly large powers of primes. The prime factorization of a number is essentially "random". That makes numbers that are large powers of primes rare; it's like rolling a die and continuously getting the same face. Now if you add two numbers, the prime factorization of the sum does not appear to be related to the prime factorization of the summands, so adding two very rare numbers and getting yet another very rare number seems unlikely. And that seems to be roughly what the abc conjecture says: it is not likely that all of a, b, and c are rare.

westoncb 8y ago
Great explanation—thanks. I think I got the upshot, though I am unclear on what function rad() is.
- edflsafoiewq 8y ago
  rad is the arbitrary function you asked about ;) rad(x) is the product of the distinct prime factors of x, ie. it drops repeated prime factors, rad(2^n)=2, etc.
- rocqua 8y ago
  rad is the radical of a number [1]. In short rad(x) is the product of all unique prime factors of x. So rad(8) = 2, rad(24) = 6, rad(75) = 15 [1] https://en.wikipedia.org/wiki/Radical_of_an_integer https://en.wikipedia.org/wiki/Radical_of_an_integer
westoncb 8y ago
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...
- edflsafoiewq 8y ago
  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 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.

westoncb 8y ago

Great explanation—thanks. I think I got the upshot, though I am unclear on what function rad() is.

edflsafoiewq 8y ago
rad is the arbitrary function you asked about ;) rad(x) is the product of the distinct prime factors of x, ie. it drops repeated prime factors, rad(2^n)=2, etc.
rocqua 8y ago
rad is the radical of a number [1]. In short rad(x) is the product of all unique prime factors of x. So rad(8) = 2, rad(24) = 6, rad(75) = 15 [1] https://en.wikipedia.org/wiki/Radical_of_an_integer https://en.wikipedia.org/wiki/Radical_of_an_integer

westoncb 8y ago

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...

edflsafoiewq 8y ago
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 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.