

Mathematicians Come Closer to Solving Goldbach's Weak Conjecture - suprgeek
http://www.scientificamerican.com/article.cfm?id=goldbachs-prime-numbers

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tokenadult
Mathematician Terence Tao, a Fields medalist and the youngest ever
International Mathematical Olympiad gold medalist to date, did the work
described in the article kindly submitted here. His blog

<http://terrytao.wordpress.com/>

is not to be missed. The "Career Advice" section of the blog

<http://terrytao.wordpress.com/career-advice/>

includes several classic articles, of which I especially like and recommend
"Does one have to be a genius to do maths?"

[http://terrytao.wordpress.com/career-advice/does-one-have-
to...](http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths/)

(Tao's answer is no.)

~~~
temphn
Don't get it. How can you celebrate someone as the "youngest ever
International Math Olympiad gold medalist" while simultaneously denying the
importance of genius in math?

Terence Tao is a humble, self-effacing guy. Or at least he knows that society
expects him to act like that rather than mimic the chest-thumping, endzone-
dancing wide receiver. So obviously he's not going to say that "yeah, to
really move math forward you kind of need to be a genius". Though of course
other geniuses like Feynman were more up front about that truth.

Very few people have the innate ability to do what Tao or Feynman did.
Recognizing that early is a mercy that allows people to move into other areas
where they have a comparative advantage, rather than beating their head
against the wall for 10,000 hours only not to get anywhere.

~~~
AjJi
But how would you know that you don't have that, supposedly, innate ability if
you don't try first? and why does it matter?

If you enjoy it, do it.

~~~
planetguy
You'll never know 'til you try? Sure, but most people get _plenty_ of chance
to try mathematics at school. And while school-level mathematics ain't the
same as research-level mathematics, enough of it uses the same bits of brain
that by the time you hit university you already know whether you're brilliant
at mathematics or not.

Terry Tao was clearly brilliant at mathematics as a child, though what nobody
knew then was that he'd go on to be _super_ -brilliant at mathematics as an
adult, rather than being one of the many disappointing prodigies.

Anyway, I'm pretty smart, but I don't take career advice from anyone as smart
as Terry Tao, just as I don't take dating advice from Scarlett Johannsen.

~~~
slowpoke
_> And while school-level mathematics ain't the same as research-level
mathematics_

School mathematics isn't even math, it's a bad joke. I recommend Lockhart's
Lament[1], I fully share his views. School 'math' destroyed every ounce of
interest I had in math, and now, as a CS student, I still have a hard time
getting rid of this attitude towards the subject even thought I know a lot
better now.

[1]: <https://www.maa.org/devlin/devlin_03_08.html>

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impendia
Something interesting about the article (and how analytic number theory is
done in general).

As mentioned in the article, it was proved over 50 years ago that every odd N
can be written as a sum of _three_ primes, provided N is sufficiently large.
The proof extends to cover five primes instead, if you like (indeed, write N =
2 + 2 + (N - 4), but in fact the five primes case is in fact easier).

Usually in such proofs the definition of "sufficiently large" is maybe like
10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far
worse, and it is typically difficult even to calculate such a fixed N, because
then you can't use big-O estimates in your calculations. Sometimes you can't
even compute such an N, it is "ineffective", see e.g. here for an example:

<http://en.wikipedia.org/wiki/Siegel_zero>

Tao's accomplishment, which was really cool, was to bring the value of N down
to the level where you could settle the small N case by brute force.

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zitterbewegung
The paper that is referenced in the article
<http://arxiv.org/pdf/1201.6656.pdf>

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etfb
Did the commenter named julianpenrod get confused about causality (the article
said "the weak version would follow if the strong were true"; he replied that
it is "not correct in stating that, if the weak conjecture is proved, then the
strong conjecture is prove[d]") or did the article get edited after he
commented?

~~~
warmfuzzykitten
It's hard to say. In either case, the result makes julianpenrod look rather
silly. More online publications should, as the New York Times does, note all
corrections to an article since its original appearance in a paragraph at the
bottom.

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gibybo
>Moreover, the larger the number, the more ways exist to split it into a sum
of two other numbers—let alone three.

I don't think that makes sense. Every large odd number has exactly 0 ways to
find its sum with two primes^, so they can't have more ways than small numbers
(which also have 0, of course). Perhaps he was speaking about the strong
conjecture, but then there would be no point to say 'let alone three', because
three is only relevant for the weak conjecture.

Am I missing something obvious, or did the author just word that
sentence/paragraph poorly?

^I am pretty sure this trivially follows from the fact that two odd numbers
always produce an even number.

~~~
bkirwi
He says _numbers_ , not _primes_ , so it's trivially correct.

Though that doesn't directly imply the conjecture being true for sufficiently
large numbers, since primes become increasingly sparse as one moves up the
number line... it seems more like an appeal to intuition than a real argument,
to me.

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mikeash
Is there any practical application to this? I'm not one of those silly people
who only cares about math if it can make a better lawnmower or whatever, but
I'm just curious as to whether this could make certain algorithms easier or
something, or if it's "just" a pure number theory result.

~~~
adeelk
The conjecture has been verified already for large enough n that it can be
applied without a real proof; a proof won’t give any new “real world”
applications. Like with Fermat’s last theorem, the real interest in a proof is
in the ideas that would be developed along the way. (For example, on the way
to proving FLT Wiles established the modularity theorem for a class of
elliptic curves, and his work was extended to a proof of the full modularity
theorem in 2001.)

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5xz41s0P8T5N
This is news from february.

~~~
oskarth
Unlike most programming, mathematical theorems usually survive for a bit
longer, so a longer time frame is justifiable.

