Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

quantamagazine.org

Post reply on HN
Print view View original

271–275 of 275 posts

Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

#271

Post by westoncb »

edflsafoiewq wrote: post #96
westoncb wrote: post #60

> “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 lookin…

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 a…

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...

Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

#272

Post by pas »

cycrutchfield wrote: post #249
cxseven wrote: post #176

Earlier quoted context omitted.

For the Kepler Conjecture, human mathematicians first found a way to show that it could be proved by checking a finite (but long) list of computer-verifiable claims. Here Scholze is saying that part of the proof simply hasn't been written, so there's nothing to verify. I think he's missing that computer verification proponents are also aware of that and are either 1. trying to motivate the camp that claims to underst…

It took like a decade to produce a computer verified proof for the Kepler Conjecture. And that was just for a relatively well-defined exhaustive search. That sounds like a molehill compared to Mochizuki’s Everest of a proof.

Methods have improved a lot since then.

The problem is that these papers are vague. The translation to computerized version would be a very different thing, it'd require a lot of creative writing on behalf of the translator.

Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

#273

Post by Angostura »

mrob wrote: post #215
Angostura wrote: post #80

Earlier quoted context omitted.

Wouldn;t it bump into the same kind of problems that Bertrand Russell's attempts bumped into with the Principia Mathematica? Namely it crashed into Gödel's incomplete theorem? Disclaimer: What I don't know bout maths would fill volumes, in both senses of the word Edit: From the down-votes, apparently not.

Human mathematicians are bound by Gödel's theorem in exactly the same way computer software is. If an informal human-based "proof" does something that would be impossible with a formal software-based proof, then it's incorrect. There is no theoretical advantage to running the calculations on a human brain.

Sorry, I didn’t intend to suggest that running a proof on wetware had advantages. Simply that running complete, from first principal proofs were doomed to failure

Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

#275

Post by edflsafoiewq »

westoncb wrote: post #271
edflsafoiewq wrote: post #96

Earlier quoted context omitted.

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 a…

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 usefu…

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.

Post reply on HN