After Prime Proof, an Unlikely Star Rises | Hacker News Reader

After Prime Proof, an Unlikely Star Rises

quantamagazine.org

85 pointsdigital5531 comments

A writeup by his doctoral advisor about his memories of Yitang Zhang: http://www.math.purdue.edu/~ttm/ZhangYt.pdf

I skimmed it, but in the intro why does he say:

  "The most natural conjecture is that the prime numbers appear randomly..
  If the twin prime conjecture is correct, then we may conclude that the
  [primes] are not constructed randomly"

why does he say a random distribution of primes would imply the twin prime conjecture is false?

My intuition is the opposite: a random distribution of primes implies the twin prime conjecture is true. Because suppose there are finite twin primes, set them aside; then test pairs of numbers above the largest of them. Since we assume the twin prime conjecture is false, the chances that random pairs above that number are both prime is exacty 0; but that would mean the primes are not distributed randomly...

I thought about this for a second, and it seems that you are right: if the primes were distributed randomly, the twim prime conjecture would be true (see below). Are there any analytic number theorists in the audience to explain what the quote in the parent post is supposed to mean and/or fix my mistakes?

The prime number theorem [1] states that the probability that a natural number N is prime is approximately 1/log(N). So, the probability that N & N+2 are both prime is about 1/log(N)^2. If they were distributed randomly, there would be about

integral(1/log(x)^2 dx) from x=2 to x=N

twin primes between 1 and N. This integral converges to infinity as N grows large[2] (wish I remembered enough caclulus to do it by hand). So, there is some N for which you'd expect to have 10^100 twin primes less than N, etc.

So it seems that if you decided by a coin toss whether a number should be prime (where you'd make it prime with probability 1/log(N)), there would be infinitely many twin primes.

Wikipedia seems to confirm it's a reasonable estimate [3].

UPDATE: By the way, the same argument implies that if primeness was decided by a coin toss, there would be infinitely many N such that N and N+1 are prime. There is, however, only one such N (namely 2). So, primeness is not decided by a coin toss, and the above is not a proof of the twin primes conjecture :).

[1]: http://en.wikipedia.org/wiki/Prime_number_theorem

[2]: http://www.wolframalpha.com/input/?i=Integrate[1%2FLog[x]^2%...]

[3]: http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93...

Another UPDATE:

I a cousin comment, Retric proves that the integral of 1/log(x)^2 diverges. The point is that log(n)^2 < n for x > 2 (you can prove this by calculus: the only maximum of the function log(x)^2/x is at x=e). So, 1/log(x)^2 > 1/x, and the integral of 1/x diverges.

The most fun way to prove the last statement is that the integral of 1/x from 1 to infinity is greater than 1/2 + 1/3 + 1/4 + 1/5 + ... > 1/2 + 1/4 + 1/4 + 1/8 + ... = 1/2 + 2 * 1/4 + 4 * 1/8 + ... = 1/2 + 1/2 + 1/2 + ....

The integral is:

li(x) - x / log(x)

Edit: It looks like you are right, the function is unbounded.

yes, very good Update. I had also thought of this and was wondering what the hell "randomly distributed" even means rigorously. I wrote the other respondent this by email (before your Update :) ) since I wasn't sure of my reasoning -- http://rifers.org/paste/show/3314

If it were truly random, then the twin prime conjecture might not be true, but would it be provable?

Is this a correct statement?

If the twin prime conjecture is correct, then we may conclude that the [prime] integers are not constructed randomly.

Another interesting tid-bit.. Looks to me that mathematics research is getting overwhelmingly complex...

For the next two semesters, I organized a seminar with five graduate students (including Yitang) on Prof. Hironaka’s monumental papers on 2 the theory of resolutions of singularities. I believed that we doubled the world population of those who had studied the papers after we finished two semesters. Prof. Grothendick once described those papers as among the most complicated thesises in the human history.

> Is this a correct statement?

Of course it is. By 'randomly' he refers to a maximum entropy ("uniform") distribution conforming Gauss' Prime Number Theorem. Since by that theorem the gap rises, the probability of twin primes would go to 0 and, crucially, the expected number of twin primes would be finite (or any finite gap really).

That's an interesting reason for Yitang's proof being so significant without actually reaching the small gap -- it shows primes violate eloquently the random distribution in some ways.

If the probably of a number being prime is ~1 / ln (N)

The probably of x and x + 2 being prime is ~ (1 / ln (N)) ^2. For large N, (ln (N)) ^2 < N so there are infinitely many twin primes. Because the Sum of 1/(f(x)) as x -> inifinity = infinity if f(x) < (x).

So, the real issue is if the number of twin primes significantly larger than (1/ln(x)) ^ 2.

Ah yes you're right I just naively assumed the gap going to infinity implied the number of small gaps has to be finite, thanks. I believe your argument argument sightly miscounts the twin primes, since you're giving a new "chance" after you had already drawn a prime/not-prime. A lower bound assuming the probabilistic model would be E[Twin Primes] > Sum ( 1/( ln(4k) ln(4k+2) ), which is still divergent so your point stands.

> so there are infinitely many twin primes

http://en.wikipedia.org/wiki/Twin_prime

Note the twin prime conjecture is not yet proven (your argument does not follow directly from PNT, but from the approximate probabilistic model that (almost surely?) displays PNT-like count).

> Because the Sum of 1/(f(x)) as x -> inifinity = infinity if f(x) < (x).

Here's a cute counterexample: x/(2*sign(sin(x))) < x, but the sum < infinity (you're missing an absolute value |f(x)| in your statement).

Yea, I was playing fast a loose. Though, for positive infinity you need 0 < f(x) < (x) not just |f(x)| < x. EX: f(x) = -0.5 * x

To expound on Retric's comment (I expounded much more in a cousin thread): I also first assumed that what you said must be true. When I tried to check with a calculation, it turned out that the opposite is the case.

Yea, my bad. I implicitly made the rookie mistake of assuming that f(x)->0 implies sum(f(x))->0, as shown by Retric :)

Yitang Zhang brings up the deep question of why does someone like him suffer in the world today. He is not showey at all but quiet and unassuming. In particular he is not un-confident but he's not showey in his confidence.

I suppose we don't value those traits.

I think it's more straightforward and less depressing than that. For most people it's simply hard to tell the difference between someone with those traits ("still waters run deep"), and someone who just doesn't have much to offer[0]. We do tend to correct those misperceptions given enough exposure and investment in the relationship, but it takes time -- and there's only so much time in a day.

[0] modulo subject X or situation Y -- we all have talents

  Still in the field of number theory, 
  I may not have only one problem to think about,

  but a couple of problems, like the distribution 
  of the zeros of the zeta functions and the L-functions.

Interesting to hear that he seems to be working on the Riemann hypothesis.

Interesting yet unsurprising. Proving the Riemann hypothesis (ie all non-trivial zeros of the zeta function are distributed along the line x = 1/2) is equivalent to proving Riemann's explicit formula for the prime counting function, which defines the distribution of prime numbers.

In the end, the Riemann hypothesis is a kind of sacred Graal for anyone interested in prime numbers.

Is there any way to view the film Counting From Infinity online?

Probably not; canistream.it comes up empty.

Good to read a story about an underdog.

>Andrew Granville “There’s no way that somebody I’ve never heard of has done this.”

What an asshole.

> “Never heard of him. Absolutely never heard of him,” said Andrew Granville, a number theorist at the University of Montreal, in Counting From Infinity. When Granville heard about the result and the techniques that Zhang used, he recalled saying, “There’s no way that somebody I’ve never heard of has done this.”

Fuck that guy.

When you are an expert in an academic field, you typically have spent many, many years reading about the problems in the field and the main ways to attack them. The same authors, will typically pop up again and again. Through conferences and collaborations one gets to know the main contributors in the field. On top of this, it is very rare for someone to break into a field with a major discovery. Normally people tend to hone their skills on more minor problems over time and obtain a deep understanding of the material. To top all of this off, academics quite regularly receive emails from crackpots claiming to have found a solution to p=np, etc.

I am not saying that Dr. Granville was right to immediately dismiss the concept of an unknown solving such a problem. I am just trying to provide some background for why he may of reacted as he did.

The problem was challenging. The solution was exceptional. So, maybe something about the person who solved the problem might be exceptional, that is, might fail to fit some common patterns?

Or, in this case, we can't call the success luck. So, for considering what is likely to be the situation or characteristics of the person with such an exceptional success, where are we to look? Are we to look at the dozens of people we do know or the thousands of other people, all of whom tried but failed to have such exceptional success?

So, we didn't see him at an AMS conference; he's not a full professor at a top university; he doesn't have a wife, 2.5 children, a 3 bedroom, two bath house, and two late model cars; ...? So?

If we hear a claim of some astounding accomplishment, then maybe (A) the accomplishment is nonsense and the person is exceptional because they are a crackpot or (B) the accomplishment is terrific and the person is exceptional because they were very successful. So, with either (A) or (B) we stand to see things exceptional that don't fit common patterns.

Net, if really want to look for the very best accomplishments, then we shouldn't reject people who look exceptional, that is, don't fit some patterns we learned from people who haven't had some terrific accomplishments.

And, in this thinking, we have to notice, in business there can be essentially some very good luck but not in pure mathematics complete with theorems and proofs that can be checked with high reliability. That is, in business, people who are not very exceptional and do fit common patterns can still be very successful because of essentially luck; so, in business can find people who fit nearly any common pattern and also are exceptionally successful just because of luck; can't do that in pure math there luck doesn't work!

Hmm there is a difference between a unknown and a crank. Zhang was a trained Ph.D. mathematician with an academic lecturing position. Now I didn't even finish my Ph.D. in physics but I know enough about how to do research to not submit a proof that is not rigorous enough to warrant peer review. One should not refuse to peer review a unknown researcher with the right credentials/trainings who make serious arguments in line with academic practices.

Same as you this is not a comment on Dr. Granville. I wrote in response to your comment on how the field is typically cloistered among a few insiders. I don't think that justifies refusing to review an unknown (as in not famous) researcher who demonstrates training and seriousness (which fortunately they didn't in Zhang's case but Zhang also was extra careful in making his ideas crystal clear). This happens too rarely for the insiders to claim undue burden. Cranks are very obvious to identify (a lack of literature review and understanding of previous works is a tell-tale sign).

Cranks and unknowns often look alike. Heck, researcher who diverge just a bit from accepted doctrines often look like cranks until either they repent or their approach bears obvious fruits.

Hmmm, that is plausible, but I can't think of any modern example where some deviation has resulted in a qualified researcher being categorized as a crank-- ignored maybe, or perhaps dismissed but definitely not called a "crank". Actual cranks are very very easy to spot and people won't bother wasting effort addressing them.

I'm not sure. Crank is often used as an ad hominem in science, so it is difficult to tell mudslinging from real crankiness, it depends on the sincerity of the one applying the label, and also on the severity of the crankiness. And many people have trouble with groking more benign eccentricity.

Are you reading this line as communicating disdain? I read it as amazement.

Generally someone isn't going to manage to become an expert at this sort of thing without talking to others within the (not terribly large!) field. That Zhang managed to do so is very unusual!

I don't think he was trying to be a dick with that quote. He expressed an honest and natural reaction: how does this monster problem get solved by someone unknown?

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