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> “The abc conjecture is a very elementary statement about multiplication and addition,” said Minhyong Kim of the University of Oxford. It’s the kind of stateme

> “The abc conjecture is a very elementary statement about multiplication and addition,” said Minhyong Kim of the University of Oxford. It’s the kind of statement, he said, where “you feel like you’re revealing some kind of very fundamental structure about number systems in general that you hadn’t seen before.” I can kind of see that, but there's something in it that gives me the feeling of arbitrariness. It's looking at the relationship between prime factors of A, B, C in A + B = C, where no prime factors may be shared between A, B, or C. The specific relationship in question is between the magnitude of C and the unique prime factors of A, B, and C multiplied together. To take an example from the article, 5 + 16 = 21 meets the basic requirements since the prime decomposition looks like (5) + (2 * 2 * 2 * 2) = (7 * 3) —no factors are shared. But, the quantity we're supposed to relate to C is (5 * 2 * 7 * 3), since it is the product only of the unique primes. Does that not feel arbitrary, to drop the repetitions? So it ends up being that the sought out smaller products arise because they had large numbers of repetitions, so we removed more when forming the product. But I suppose there are probably just some deep number theory mysteries behind it which give justification, so it only feels arbitrary to someone like myself who is basically ignorant on the subject :)

jetrink 8y ago
> Does that not feel arbitrary, to drop the repetitions? If the conjecture turns out to be true, then that apparently arbitrary step turns out to be meaningful, because it uncovers a hidden relationship between the numbers A, B and C and their factors.
edflsafoiewq 8y ago
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.

jetrink 8y ago

> Does that not feel arbitrary, to drop the repetitions? If the conjecture turns out to be true, then that apparently arbitrary step turns out to be meaningful, because it uncovers a hidden relationship between the numbers A, B and C and their factors.

edflsafoiewq 8y ago

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.