
Sudden Progress on Prime Number Problem Has Mathematicians Buzzing - lelf
http://www.wired.com/wiredscience/2013/11/prime/all/
======
hawkharris
When I ordered a set of three foot-long chicken parmesan sandwiches from
Subway a few weeks ago, the guy behind the counter said, "Because prime
numbers are fundamentally connected with multiplication, understanding their
additive properties can be tricky."

I was confused at the time, but it makes sense now.

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throwaway1979
Two points:

\- In math, an innovative "attack" (such as a technique to approach a problem)
can result in years of subsequent research that progresses a field. Zhang
deserves a lot of credit for his accomplishment!

\- He shouldn't have had to work at subway. If only our society held
intelligent people in the same regard as athletes. What we have right now
(hoops to get tenure, publish or perish culture, etc.) is a joke. I review
papers for some notable conferences in my subfield in CS. This week I read
about a dozen papers. The ones that pained me the most were written by very
smart people solving made up and artificial problems. But hey ... you need to
publish this crap to graduate/get tenure/get next grant/whatever. How about a
basic income for everybody with a PhD?

~~~
JayNYC
"How about a basic income for everybody with a PhD?" How about a basic income
for everybody?

I am always amazed, how unsuccessful PhDs in life are. Myself included.

~~~
charlieflowers
Why? Existential crisis?

~~~
dsr_
A doctorate in most fields is proof of a degree of general smartness and an
ability to work rigorously in the field. It doesn't necessarily have anything
to do with good judgement, luck, diplomacy, social skills, practical skills,
or any of the thousand other things that contribute to success in life.

~~~
jl6
To do a PhD you have to be smart enough to be _able to_ do one, and dumb
enough to _actually_ do one.

~~~
wolfgke
In my opinion rather smart and _despaired_ enough to actually do one.

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chrisfarms
> There are a lot of chances in your career, but the important thing is to
> keep thinking

I hope this quote gets picked up and cited throughout future history

~~~
themodelplumber
What is it about this quote that makes you feel it should be cited throughout
future history? Just curious.

~~~
coldtea
It seems some people think being reminded to think is profound.

------
taspeotis
Previous discussion here:
[https://news.ycombinator.com/item?id=6766097](https://news.ycombinator.com/item?id=6766097)

~~~
z92

        Previous discussion here

272 points, 93 comments, 3 days ago.

------
abuehrle
I'm confused by the "Admissible Combs" sidebar. Can someone help me
understand?

"Roughly speaking, a comb is admissible if there is no obvious reason why its
teeth couldn’t point entirely to primes infinitely often as you move it along
the number line"

Then: "A much more audacious conjecture called the prime k-tuple conjecture
... posits that any admissible comb will point entirely to primes infinitely
often."

Isn't this just saying the prime-tuple conjecture states that admissible combs
are admissible?

~~~
justinpombrio
Admissibility doesn't say that a comb's teeth _will_ point entirely to primes
infinitely often, it merely says (as you quoted) that there's no _obvious
reason_ its teeth _won 't_ point entirely to primes infinitely often.

I suspect that "obvious reason" is being used as shorthand for some technical
definition the article doesn't try to state. Any mathematician here that knows
more?

~~~
impendia
Sure. {n, n + 2, n + 4} is a 3-tuple that is _not_ admissible: One of n, n +
2, and n + 4 will always be divisible by three, so the tuple has no chance to
ever represent three prime numbers for one value of n. (Except for n = 3.)

Conversely, {n, n + 2, n + 6} is an admissible 3-tuple. There's no fixed
integer that will always divide at least one of n, n + 2, and n + 6, so we
believe that there are infinitely many n for which n, n + 2, and n + 6 are all
prime.

~~~
justinpombrio
And so the claim is that there is no less obvious reason that a comb might not
be all primes only finitely often. That _is_ an audacious claim :-).

------
jostmey
Not being able to understand the intricacies of this proof, I can take away
but one message. Genius can come from anywhere, even from an academic who was
only able to find a job working at a Subway diner. So be open to good ideas -
they can come from anybody, anywhere.

~~~
jostmey
And to top it all off, this Mathematician was in his late fifties when he
published his breakthrough proof!

~~~
eps
It's not the age, it's the marital status that affects the productivity the
most.

~~~
tomrod
I find myself downright inspired as a family man.

~~~
fhars
You should feel depressed instead. In men, creativity is destroyed by marriage
[http://www.abc.net.au/science/articles/2003/07/11/900147.htm](http://www.abc.net.au/science/articles/2003/07/11/900147.htm)

~~~
tomrod
I'm very appreciative of the the statistical concept of outliers.

As Hume would point out, is is not ought.

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Xeoncross
Does this mean as much as I think it does? Aren't most of our current
cryptography schemes based on prime numbers taking time to calculate?

~~~
csense
No.

Finding random primes is easy, especially if you don't mind using methods
which allow an extremely tiny probability of error [1] -- this is standard
practice in cryptography.

Finding random big primes p and q is easy. Multiplying them together into a
product r = pq is easy. RSA cryptosystem picks p and q when generating the
key, multiplies them, then throws away original p and q values. Only r is
stored in the key.

Cracking the RSA algorithm itself [2] is basically equivalent to turning r
back into p and q.

RSA is based on the hardness of factoring an integer into a product of primes.
I don't think this conjecture will lead to practical improvements in
cryptography / cryptanalysis, at least not with respect to cryptosystems that
are currently widely used.

[1]
[http://en.wikipedia.org/wiki/Primality_testing#Probabilistic...](http://en.wikipedia.org/wiki/Primality_testing#Probabilistic_tests)

[2] Of course, there are many more attack vectors on specific
_implementations_ of the RSA algorithm, like trying to gain information on the
secret key by measuring time taken to encrypt a message, measuring heat/power
consumption/electronic noise, taking advantage of implementations that use a
poor source of randomness, hacking into the system where the private key is
stored, or bribing/threatening a human with legitimate access to the private
key (as covered in [http://xkcd.com/538/](http://xkcd.com/538/)).

~~~
Someone
_" Finding random big primes p and q is easy"_

For those wondering why then, it is news when a new largest prime is found:
there are gradations of 'big'. The big primes used in cryptography have fewer
than 1000 bits, or about 300 digits. That is not what one (nowadays) calls
'large' in the field of finding the largest prime. There, one laughs at
numbers of a million digits (the largest known prime has 17,425,170 digits)

~~~
csense
Also, for a practical engineering problem, like writing an RSA implementation
which will be used in an SSL implementation which will encrypt credit card
numbers on an e-commerce website, it is okay to use probabilistic tests which
have a tiny chance of a false positive (the test claims a number is prime when
it is actually composite).

You can generally repeat a probabilistic test with different parameters to
make the probability of a "false positive" prime number very tiny -- like .000
000 000 001 -- while still staying within the computational budget manageable
on an ordinary computer (obviously, a practical Internet security scheme
should be able to complete its computations in a second or two on a typical
desktop, laptop, server, and smartphone).

So probabilistic tests are good enough for practical purposes, but you need a
non-probabilistic test to get the false positive probability all the way down
to zero. And you need to use a method that has zero percent false positive
chance to claim a world record. The EFF-administered monetary prize for a
record-breaking prime number, for example, requires "the primality proof must
be a deterministic proof..." [1] So you can't use probabilistic methods.

[1]
[https://www.eff.org/awards/coop/rules](https://www.eff.org/awards/coop/rules)

~~~
mistercow
I know (or think) that most record primes these days are Mersenne primes due
to the relative ease of testing their primality. What I wonder is if
probablistic tests could be used to find candidates for very large non-
Mersenne primes before testing them using AKS.

Or is o̅(log⁷·⁵(n)) just infeasible for those sizes of numbers?

------
coldcode
Since counting numbers are so regular I've always wondered why finding primes
is so hard. Then again I have no hair so using combs is a mystery.

------
Houshalter
Related questions, what are the open problems with prime numbers? Not that I'd
ever solve them, but I was having fun using machine learning algorithms try to
predict the next prime number in a sequence and it was kind of interesting.

~~~
wolfgke
\- Goldbach's conjecture
([http://en.wikipedia.org/wiki/Goldbach%27s_conjecture](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture))

\- Goldbach's weak conjecture
([http://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture](http://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture))
- Harald Helfgott claims that he proved it

\- Is factoring in P? (it is known that prime testing is in P; thus the
factoring is in NP ∪ coNP) - if it were true, RSA would be broken immediately
(and some expert that I talked to told me, he believes that such a factoring
algorithm could probably be extended to break ECC, too)

\- Riemann hypothesis
([http://en.wikipedia.org/wiki/Riemann_hypothesis](http://en.wikipedia.org/wiki/Riemann_hypothesis))

\- Do for each even number n exist two prime numbers p > q such that n = p-q?

------
danielharan
What I find surprising is that there's an online tool for math collaboration
(github for maths?), and that it turns out to be rather useful.

------
charlieflowers
Are they still at 600?

~~~
enum
This abstract suggests that the bound may be lower:

[http://jointmathematicsmeetings.org/amsmtgs/2160_abstracts/1...](http://jointmathematicsmeetings.org/amsmtgs/2160_abstracts/1096-11-441.pdf)

From
[http://jointmathematicsmeetings.org/meetings/national/jmm201...](http://jointmathematicsmeetings.org/meetings/national/jmm2014/2160_program_ss23.html)

~~~
fhars
No. That abstract says that the bound it 12 if the Elliot-Halberstam
conjecture holds. Under that assumption, the bound has been 16 since 2000. The
trick that brings the number down from 16 to 12 under that assumption is the
same that brings the proven number down to 600. This is actually described in
the linked wired article.

~~~
charlieflowers
Ah. Thanks for the clarification.

------
fractalsea
Current list of attempts:

[http://michaelnielsen.org/polymath1/index.php?title=Bounded_...](http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes)

------
dmak
Could someone explain this in layman's terms? What does it mean to have
infinitely many pairs of primes separated by 600?

~~~
lutusp
If the original claim is true, it means that the maximum numerical separation
between primes is 600, meaning primes may sometimes be separated by a smaller
gap, but no two primes will be separated by a larger gap, a numerical distance
greater than 600. What this means is, if there are no gaps between primes _a_
and _b_ greater than 600 (i.e. |a-b| > 600), if all prime pairs have a smaller
gap, then given the size of the set under consideration, it follows that there
are an infinite number of pairings {a,b} in which their separation is 600 or
less (|a-b| <= 600).

Also, in mathematics, finding an example of a property isn't very difficult.
The difficulty lies in proving a theorem that makes a universal statement
about that property -- that's the real challenge.

If that answer wasn't clear, don't hesitate to ask a clearer question.

~~~
ipince
I don't think this is right.

There are certainly prime gaps that exceed 600. In fact, the higher you up (in
the number line), the higher these prime gaps tend to get.

[http://en.wikipedia.org/wiki/Prime_gap](http://en.wikipedia.org/wiki/Prime_gap)

What's being said here is that despite of that, there will always be prime
gaps smaller than 600. No matter how high you go, you can always find a pair
of primes that are separated by less than 600. In other words, pick the
biggest N you can ever dream of, and there will exist primes p1 and p2 such
that they are both bigger than N and their difference is smaller than 600.

~~~
lutusp
Yes, you're right, and it's too late for me to either delete or edit my
original post. :(

In light of that, I think the discussion and work revolves around discovering
the smallest gap as the numbers themselves become larger and tend toward
infinity. Obviously the very smallest prime gap is that between 2 and 3, i.e.
1, and there are a great number of primes separated by 2, and the Twin Prime
Conjecture asserts that there will always be occasional pairs of primes
separated by 2 no matter how large the numbers themselves become. The present
work is in part meant to put that conjecture on a more analytical footing.

I would love to delete my original post.

