
Mathematicians Team Up on Twin Primes Conjecture - digital55
https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/
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davidjohnstone
To clarify, this result means that there are infinitely many pairs of primes
separated by at most 600. This doesn't mean that the gap between subsequent
primes must always be less than 600.

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raz32dust
Ah I had missed it too. Is there a bound on the gap between subsequent primes
as well?

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xyzzyz
For any natural n, the n consecutive numbers (n+1)! + 2, (n+1)! + 3, ...
(n+1)! + (n+1) are all composite -- the first one is divisible by 2, the
second one by three and so on.

Thus there are arbitrarily long sequences of consecutive numbers that have no
primes in them.

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eru
I seem to remember that there's always a prime between n and 2*n? (Still an
arbitrarily large absolute gap, but a fixed relative gap.)

Can anyone deny or confirm?

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DerekL
Yes, that's called Bertrand's postulate, first proven in 1850 by Chebyshev.
For any integer n > 3, there's at least one prime strictly between n and 2n-2.

[http://en.wikipedia.org/wiki/Bertrand's_postulate](http://en.wikipedia.org/wiki/Bertrand's_postulate)

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joe_the_user
Ah, an earth-shaking discovery made by a man in his fifties who was barely
within academia [1]. Guess he won't get a Fields Medal for that.

It seems like more and more of the really big discoveries are being made by
people who go off and work really hard on their own.

I vaguely recall a claim that the greatest discoveries aren't being made by
people in their twenties any more just because math is so huge you need ten
years of study to get to the bleeding edge.

Anyway, it seems to say something about academia.

Edit: Just to be clear I neither deny that academia can achieve great things
nor do I claim that young people can't do similarly. Certainly, anyone halfway
near academia probably see many flaws but that's for another post.

[1]
[http://en.wikipedia.org/wiki/Yitang_Zhang](http://en.wikipedia.org/wiki/Yitang_Zhang)

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icambron
I don't get it. The article talks not just about Zhang, but also about James
Maynard, who is a postdoc and looks young, and is part of what you might call
standard academia. His result is independent of Zhang's and is orders of
magnitude stronger and is much easier to understand. Comes off pretty good for
academia, really.

Also, remember that most people on Earth aren't in academia, but most people
still think. So the null hypothesis is that most innovations won't come from
academia. But a _huge_ amount of them do anyway, including even your
counterexample, who--despite his economic struggles--was still a lecturer at a
university. You could make a similar argument for young people.

Our university system and our cultural obsession with youth deserve plenty of
criticism, but the accusation that they're not making important discoveries
isn't one of them, and it's totally unwarranted from this article.

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azakai
> The article talks not just about Zhang, but also about James Maynard, who is
> a postdoc and looks young, and is part of what you might call standard
> academia. His result is independent of Zhang's and is orders of magnitude
> stronger and is much easier to understand.

Maynard has done some stellar work to be sure, but he built off of Zhang's
breakthrough. Maynard's result is one of a sequence of very quick findings all
happening in just months after Zhang's work became public, all make possible
by that work, and relatively easy, as can be seen from their number and rapid
pace.

I have not read all the papers here, but my guess is that Zhang's result is
hugely more impressive.

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callmecosmas
Actually at the bottom of page 2 of Maynard's preprint he states: "We
emphasize that the above result does not incorporate any of the technology
used by Zhang to establish the existence of bounded gaps between primes."

So it seems like it is actually a coincidence, though they were both building
off of the same previous work (GPY) so maybe it's not that surprising.

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ww2
That is only a tactic move. it is Zhang's work inspired him to look for a
quick better alternative. Zhang showed " there is gold! " And then Maynard ran
towards that.

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jtchang
So if I understand this right...it means that no matter how far along the
number line I go (and I can go a long ways out) I will always be able to
eventually find a pair of primes that are separated by no more than 600?

The crazy thing to me is that primes get less and less dense as you get
further and further out. It would seem to me that once you reach a certain
threshold you won't be able to find two primes "close" to each other anymore
because primes will be so far apart from each other. But this says I will
always be able to find a pair pretty close together? (I may be looking a long
time though)

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abuzzooz
> It would seem to me that once you reach a certain threshold you won't be
> able to find two primes "close" to each other anymore because primes will be
> so far apart from each other. ]

That is not quite correct. The _average gap_ between primes gets larger as you
move further along the number line. But, you will still get the occasional
consecutive odd numbers that are close to one another. In fact, there is an
old conjecture (Twin Prime Conjecture[1]) that claims that there are an
infinite number of pairs of consecutive primes separated by 2.

PS. I'm not a mathematician (engineer with some decent math background), so I
might very well be wrong.

[1]
[http://en.wikipedia.org/wiki/Twin_prime](http://en.wikipedia.org/wiki/Twin_prime)

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fragsworth
Yeah, this result is effectively getting a little bit closer to proving the
Twin Prime Conjecture.

It would be interesting to see the proof get progressively closer to 2, but
more and more difficult each time.

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simonista
Wow, this is a great story. First of all, props to the writer for weaving a
bunch of different threads together in a way that made sense and was
understandable to non-mathematicians without dumbing it down too much. And
this is why I love math. There are so many great characters and themes in this
story.

I love problems that can be stated so simply and yet are so fiendishly
difficult to prove.

I love the idea that there is room for everybody in mathematics, from the lone
genius outside of academia, to the young post-doc, to the unlikely
collaborators.

Most of all, I love the idea that it's okay to fail, to throw out dumb ideas
along with brilliant ones and that everyone makes mistakes, even Ph.D.'s in
mathematics. We need more of that in education and academia generally.

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whatgoodisaroad
My understanding of the Riemann Hypothesis is that it can be reformulated as a
statement about prime density. If this result is a concrete outer bound on
density, does it have implications on RH?

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tzs
Speaking of reformulations of the Riemann Hypothesis, there is a surprisingly
elementary statement that is equivalent to the RH.

Let H(1) = 1.

Let H(2) = 1 + 1/2.

Let H(3) = 1 + 1/2 + 1/3.

...and so on.

These are called the harmonic numbers.

Let S(1) = 1.

Let S(2) = 1 + 2.

Let S(3) = 1 + 3.

Let S(4) = 1 + 2 + 4.

Let S(5) = 1 + 5.

Let S(6) = 1 + 2 + 3 + 6.

The pattern here is harder to spot than the pattern for the harmonic numbers.
S(n) for a positive integer n is the sum of the divisors of n. For instance, 6
is divisible be 1, 2, 3, and 6, so S(6) = 1 + 2 + 3 + 6.

Conjecture: for any positive integer n, S(n) <= H(n) + exp(H(n)) log(H(n)),
with equality only when n = 1.

Believe it or not, this conjecture is equivalent to the Riemann Hypothesis!

Proof here:
[http://xxx.lanl.gov/abs/math.NT/0008177/](http://xxx.lanl.gov/abs/math.NT/0008177/)

~~~
whatgoodisaroad
That is surprising. It's pretty incredible to see a formulation that doesn't
involve any advanced Number Theory.

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adamfeldman
That is the most beautifully well-written article about mathematics that I
have read in my life.

The explanations made it completely accessible as a CS student with nothing
more than calculus experience and familiarity with the concept of the search
for primes.

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mrcactu5
I knew about the Wiki
[http://michaelnielsen.org/polymath1/index.php?title=Bounded_...](http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes)

However, I did not know about the solo paper
[http://arxiv.org/abs/1311.4600](http://arxiv.org/abs/1311.4600)

    
    
      In their celebrated paper [5], Goldston,
      Pintz and Yıldırım introduced a new method for counting   
      tuples of primes, and this allowed
      them to show that [eq 1.1]
    
      The recent breakthrough of Zhang [9] managed 
      to extend this work to prove [eq 1.2]
    

"Sieve methods" say integer properties like divisibility are essentially
random. This is how we can show two integer randomly chosen are relativley
prime with probabilty pi²/6= 0.608...

[http://math.stackexchange.com/questions/64498/probability-
th...](http://math.stackexchange.com/questions/64498/probability-that-two-
random-numbers-are-coprime)

Here we are trying to count primes separated by certain small gaps.

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Maven911
Montreal..on the map, finally. There's a lot of talent here.

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jrockway
Montreal was already on the map for Bixi. And probably other things.

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possibilistic
I'm not a mathematician, but this is an interesting subject to me.

Does this imply a certain density to prime numbers? And given this new
information, does it mean that it could be less computationally hard to find
or verify primes?

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iclelland
The overall density of primes is known, and decreases continuously as the
numbers get larger (See
[http://en.wikipedia.org/wiki/Prime_number_theorem](http://en.wikipedia.org/wiki/Prime_number_theorem))

What this theorem says is that no matter how large the numbers get, no matter
how sparsely the primes are spread out, _on average_ , there will always be
pairs of primes that are relatively close (within 600 of each other)

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ars
"on average" is the wrong concept here. Just leave it out of your sentence and
it's correct.

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thedufer
That's what I thought, but if you apply "on average" to the clause preceding
it rather than the one following it, it is correct.

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drakaal
The bigger implication is that if you know the frequency of prime numbers and
have a way to calculate when to expect them the difficulty in calculating new
large primes is greatly reduced. In effect the strength of all encryption
based on Primes is greatly reduced.

BitCoin, PGP, HTTPS, Certs, all just got orders of magnitude easier to crack.
Well not just... It will take a little bit for time to use this information to
create optimizations for beating each of these. But soon.

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velis_vel
What? No. RSA doesn't depend on finding prime numbers, it depends on factoring
large numbers, which is a completely separate problem. Elliptic curve
cryptography doesn't depend on prime factorization at all.

Even if they did, I don't think this really helps you; knowing that there
_exist_ arbitrarily many primes within N doesn't help you find primes any
faster.

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drakaal
What do you think Factors are? Do you know what a Prime is? Do you know what a
Factor Is?

I am skipping the rest of your arguments since. "What, Yes" is more correct.

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sushirain
A mathematician once told me that he doesn't appreciate research which leads
to small improvement of a bound, but only research that leads to a conceptual
novelty (like possibility). I guess this story shows that there is room for
both.

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misiti3780
I might be way off here ... but does this not have implications for RSA
encryption .. isnt the whole concept resting on the fact that two prime
numbers cant be factored ... (I don't know shit about encryption obviously)

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gizmo686
It has not actually been proven that RSA is backed by the difficulty of
factoring. That is to say, it may be possible to break RSA without an
efficient way to factor numbers. However, an efficient way to factor numbers
does break RSA.

Having said that, there is no obvious way that this result helps with
factoring. Indeed, the Twin Prime Conjecture states that the gap is actually
2. Obviously, this has not been proven, but if there was a way to use a proof
of this fact to efficiently factor numbers, then we would have used that
method without the proof. If it ends up not working, then we would have
disproven the Twin Prime Conjecture.

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ww2
Maynard is too eager. He talked too much that is a coincidence. But the fact
is that he is a member of the polymath project and all the works are
consequences rather than coincidence.

