

Unheralded Mathematician Bridges the Prime Gap - nature24
https://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

======
6ren
> Without communicating with the field’s experts, Zhang started thinking about
> the problem. After three years, however, he had made no progress. “I was so
> tired,” he said.

> To take a break, Zhang visited a friend in Colorado last summer. There, on
> July 3, during a half-hour lull in his friend’s backyard before leaving for
> a concert, the solution suddenly came to him. “I immediately realized that
> it would work,” he said.

<http://c2.com/cgi/wiki?FeynmanAlgorithm>

    
    
      Write down the problem.
      Think real hard.
      Write down the solution.

~~~
ma2rten
Been there, done that.

It is not as glorious as it sounds. I had some very strong intuitions about
some obscure computer science problem (maybe 10 people worked on it worldwide,
but I think they gave up already). I was thinking about this problem for at
least one year, with some periods of very intensive thinking. I could have
really used that time more productively, because that was already quite a busy
period for me, but the solution seemed so close.

After about one year, I finally had a solution mapped out, which I think would
work. However, there were some major implementation problems (memory
complexity). That was the time when I was finally able to let it go.

So the advise that I would give to people is: focus on real-world problem,
that real people have.

~~~
claudius
> focus on real-world problem, that real people have.

Because the world would be a much better place today if we hadn’t bothered
about quantum mechanics, special and general relativity or even advanced
mathematics to handle these topics. Sure, this sort of thing won’t solve the
problems of this specific time period, but it might well be relevant to a
future generation’s problems.

~~~
bzbarsky
Quantum mechanics solved a bunch of real world problems that were relevant at
the time. As one example, people were trying to predict heat capacities of
gases (for things like chemical engineering industrial processes) and the
theory they had, based on equipartition, produced results that didn't match
experiment at room temperature. And we're not talking a few percent mismatch;
we're talking significant differences.

Special relativity likewise addressed specific experimental problems people
were having.

General relativity you may have more of a case for.

------
Xcelerate
So, there's two (slightly off topic) questions this article raises that I've
had for a while. Perhaps someone who's been in academia longer than I have can
enlighten me.

First, the article talks about experts within the field. I've been a graduate
student for about a year now, and within my field of research (molecular
dynamics), I still have no idea who the "experts" are. I see the same names
frequently pop up on papers I read -- are these the experts? I don't know of
any online forums where people discuss MD; if they're out there, then they
must be a secret. I'm sure I can't just email these professors and say "hey,
want to chat?" So where does this group collaboration and unanimous
identification of who the leaders in a field are come from? Conferences are
only a few times a year; I can't imagine that's where all of these people meet
up and socialize.

And secondly, how is this mathematician "virtually unknown", as the article
puts it? The University of New Hampshire is surely a well-known institution.
I'm sure it ranks top 100-500 in the world, right? And there's maybe 20-25
mathematics professors at each university. I also know math is very diverse,
and an expert in topology isn't going to know too much about number theory. So
within each "domain" of math, there can surely only be a few hundred in the
world at any given time who are actively researching the subject. How can he
be so unknown then?

~~~
impendia
I am a professional mathematician and number theorist, who has studied work
closely related to Zhang's.

I had _never_ heard before of an older (40+) mathematician, who has done
essentially no meaningful work in the subject before, and has virtually no
publication record, coming out of seemingly nowhere and proving such a big
theorem.

This is in no way a slight against older mathematicians, indeed many
spectacular results are proved by people over 40, but they typically
accumulate an excellent track record on the way.

Indeed, I fully assumed that this guy was full of shit. However, I have heard
that well-known experts in the area have closely read Zhang's paper and found
it to be correct.

This is enough to put a smile on my face. There is something wonderful about
skepticism and cynicism being proven wrong, especially when the skepticism is
my own.

~~~
Xcelerate
Ah, I did not realize he had not really published before. I suppose that is
very surprising. Also, the way your post is written seems to suggest that most
results out of nowhere are "crankish". Are there really people who spend their
time working on these things that aren't professional mathematicians and who
submit flawed papers to math journals? I would think that would be an
unproductive use of one's time...

~~~
impendia
> Are there really people who spend their time working on these things that
> aren't professional mathematicians and who submit flawed papers to math
> journals?

Yes, as well as people who are professional mathematicians (usually not well
known) who also do so.

There is not a flood of such papers, but my thesis advisor, in his capacity as
editor of the Proceedings of the American Mathematical Society, gets at least
a dozen or so such papers each year. I refereed some of them, and had to
explain to these authors why their proofs were mistaken.

~~~
cantos
Attempted proofs of Fermat's Last theorem by cranks used to be so common that
Edmund Landau, a German mathematician, had a form letter for them: “Dear
Sir/Madam: Your proof of Fermat’s Last Theorem has been received. The first
mistake is on page _____, line _____.”

I have sometimes seen notes on the pages of prominent mathematicians that they
don't have time to examine unsolicited attempts at major problems by amateurs
so there must still be a good number. I also have known someone whose sibling
is a software engineer and whose hobby is trying to resolve P ? NP.

~~~
duaneb
I heard a talk by one of the main vetters of the P?NP problem (which itself
was underwhelming; clearly presentations were not his strong suit) and he said
there were vast numbers of attempted proofs, only a small fraction of which
are even treated seriously, and none of which are considered for long enough
to be considered "close" to complete. He seemed to believe that the only
significant hope for the problem was in the hands of Ketan Mulmuley at
UChicago (<http://www.cs.uchicago.edu/people/mulmuley>), but realistically a
solution would not be found in our lifetimes.

This is not only interesting for that specific problem, but it was revealing
in how many amateurs (including those with doctorates) who are completely over
their head at the edge of our knowledge—the people who understand it well
enough to evaluate an approach adequately don't submit because they understand
the difficulty in ways most don't.

------
banachtarski
This is just utterly inspiring and something I've always dreamed about
happening to me. Too often, people are proud of themselves after figuring out
something that took a few days or even a few weeks to come up with. To stumble
upon a solution when you had all but given up hope after three YEARS is just
astounding and must amazing.

Congratulations to this mathematician. Inspiring. Truly inspiring.

~~~
ky3
_three YEARS_

Persistence, man, persistence.

The chinese have some saying that, loosely translated, "ten years is short to
revenge a father's death."

~~~
est
君子报仇十年不晚

Also this:

10 minutes on stage is 10 years practice behind stage.

~~~
ky3
Neato, thanks!

[http://en.wiktionary.org/wiki/%E5%90%9B%E5%AD%90%E6%8A%A5%E4...](http://en.wiktionary.org/wiki/%E5%90%9B%E5%AD%90%E6%8A%A5%E4%BB%87%EF%BC%8C%E5%8D%81%E5%B9%B4%E4%B8%8D%E6%99%9A)

"A nobleman bides his time for the perfect moment for revenge."

I swear I didn't make up the oedipalia! A chinese native mentioned it in
passing years ago and the phrase stuck.

------
nirvanatikku
This is simply amazing. Stories like this are humbling, inspiring and provide
a renewed appreciation for persistence.

Also, with respect to being unknown, the one thing that does pop up upon doing
a search is his rating on ratemyprofessor:
<http://www.ratemyprofessors.com/ShowRatings.jsp?tid=56169>.

~~~
aarondf
The common refrains seem to be 1) easy tests 2) seems nice in class and 3) is
mean one in one.

I wonder if 2 & 3 have anything to do with his shyness (as mentioned in the
article) and his lack thereof when he gets to talking about math? Maybe he
gets on a roll as far as math goes, but one-on-one interactions are more
difficult?

Or maybe he was busy coming up with this bad-ass paper and didn't want to
extend his office hours?

Either/or.

------
RockofStrength
The Twin Prime conjecture is probably the most elementary unsolved mystery in
math.

------
pjdorrell
Is the bound achieved by his proof actually 70000000? That looks like a
suspiciously round number. Why can't they tell us the exact bound that he
achieved? (Presumably some slightly smaller number?)

~~~
vlasev
In math you often don't need super precise bounds. I think it just happens
that 70m works. If you try harder you can probably find something more
accurate. IMO even a bound like 100m is good because it's a finite number.

~~~
pjdorrell
No, actually any proof that supplies a bound must also have some least bound
that can be derived from that proof. It could be a calculation that starts
from the bound, or it could be some calculation with a result that tells you
what the bound is. To me it seems very unlikely that the actual bound was
exactly 70000000 in either case.

It _could_ be that some super-expensive calculation is required for each
candidate bound, and Zhang guessed 70000000 as a starting point and did the
calculation for that number, which succeeded, but he didn't bother repeating
it for any smaller numbers because it would cost him too much. But if that was
the case, that would be interesting in itself.

------
ColinWright
Also reported six days ago, although with comparatively little discussion:

<https://news.ycombinator.com/item?id=5703219>

------
chime
> Among large numbers, the expected gap between prime numbers is approximately
> 2.3 times the number of digits; so, for example, among 100-digit numbers,
> the expected gap between primes is about 230.

> no matter how sparse the primes become — you will keep finding prime pairs
> that differ by less than 70 million.

Does this mean even for numbers longer than 70m/2.3 = 31m digits, there is
bound to be at least one prime every 70m numbers or so?

~~~
jckt
Edit: Admittedly only skimmed the article, let alone the paper. Obviously
there are an infinite number of primes and we've proved that ages ago. Do not
read this comment, read the replies if you want a more accurate answer.
Leaving it here for posterity.

Yup, at least according to the article:

"His paper shows that there is some number N smaller than 70 million such that
there are infinitely many pairs of primes that differ by N. [...] [N]o matter
how sparse the primes become — you will keep finding prime pairs that differ
by less than 70 million."

So he proved that it there is some number N, not necessarily 70mil but below
70mil, is the "gap" between primes.

\--------

This is an amazing development. On the other hand, we are also getting close
to nailing the odd Goldbach Conjecture. I think it was just last year that Tao
proved, without the Riemann Hypothesis, that every odd number N > 1 can be
expressed at by the sum at most 5 primes. Wonderful to witness such great
leaps in maths during one's own lifetime.

~~~
heretoo
No, this isn't right.

He showed that there are infinitely many twin primes differing by 70 million
from each. However, it could be that the next such twin prime, has its lower
prime number more than 70million numbers further along.

What I mean is, there is allowed to be a gap larger than 70million with no
primes in it, then all of a sudden, twin primes. But those twin primes with a
gap less than 70million between them are guaranteed.

Basically, if you separate all adjacent pairs of primes, with no primes in
between, you can separate this into two groups of those adjacent pairs with a
gap less or equal to 70mil, or greater than 70mil. The first group has
infinitely many members according to the proof. And the second group is
probably not empty.

~~~
ky3
_And the second group is probably not empty._

We know _for sure_ it's non-empty by simple factorial arguments ITT or by
appeal to the prime number theorem.

------
seanlinmt
Wondering if this impacts the security of public key encryption.

~~~
JackGibbs
As part of properly generating RSA keys there are restrictions as to how close
together the two primes numbers chosen can be. So no, twin primes bear no
relevance to the problem.

[http://en.wikipedia.org/wiki/RSA_(algorithm)#Faulty_key_gene...](http://en.wikipedia.org/wiki/RSA_\(algorithm\)#Faulty_key_generation)

