
Every odd number greater than five is the sum of three primes - ColinWright
https://plus.google.com/114134834346472219368/posts/8qpSYNZFbzC
======
CurtMonash
One of the amazing things about mathematics is that you'd like to work in one
branch and wind up working in another. E.g., who knew before Riemann that
number theory would often wind up being an exercise in calculus of one complex
variable?

The example perhaps best known to non-mathematicians is that there were two
formulations of quantum mechanics, namely in group theory and in partial
differential equations, and they turned out to be pretty much the same thing.

\--------------------------------

Come to think of it, that's been a great lesson for me in business life. I
come in supposedly to consult about one kind of problem, and really wind up
addressing a different subject entirely.

~~~
chii
i find it more amazing that some seemingly completely unrelated things are
somehow deeply connected (and unintuitive connected), and in reality are two
facets of the "same object".

~~~
X4
It appears like the constant flux that the new discoveries provoke fruit in
better and better solutions. This is utter joy, the heist for more knowledge.
Evolutionary and peer reviewed improvments :)

------
danbruc
It is absolutely mind-blowing how much time and passion humans have invested
in understanding the consequences of nine simple statements...and how hard
they resist against unveiling their secrets.

    
    
      0∈ℕ
      ∀x∈ℕ:S(x)∈ℕ
      ∀x∈ℕ:S(x)≠0
      ∀x,y∈ℕ:S(x)=S(y) ⇒ x=y
      0∈X∧∀x∈ℕ:(x∈X ⇒ S(x)∈X) ⇒ ℕ⊆X
      x + 0 := x
      x + S(y) := S(x + y)
      x · 0 := 0
      x · S(y) := x · y + x
    
    

These are the Peano axioms [1] defining the natural numbers with addition and
multiplication. And for at least 3000 years humans are tying to figure out
properties of these natural numbers, all consequences of these definition, and
the end is not in sight.

[1] <https://en.wikipedia.org/wiki/Peano_axioms>

~~~
cscurmudgeon
I wouldn't say they define the natural numbers. A more apt phrasing would be
"describe the natural numbers." They are more like the laws of physics for the
natural numbers.

Also, there are easily understandable theorems about the natural numbers not
provable from the Peano axioms.

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

~~~
strangestchild
How then are the naturals defined? By the Von Neumann construction? (0, {0},
{0, {0}}, ...)

~~~
cscurmudgeon
I would say that the Von Neumann construction and PA try to capture the
natural numbers. It is a philosophical stance which can be called platonism.
What you say is called the formalist stance.

~~~
strangestchild
What I mean is that since Goodstein's theorem is provably true for the
naturals, but is not a consequence of the Peano axioms, then the definition of
the naturals used to demonstrate Goodstein's must be strictly stronger than
the Peano axioms themselves. I was wondering what this definition might be.

I'm familiar with the distinction between formalism and Platonism, although I
still haven't made my mind up yet :)

~~~
danbruc
It is a consequence of the Peano arithmetic (Peano axioms plus definition of
addition and multiplication), it is just not provable in this system.

Gödel's first incompleteness theorem [1] states this fact, that no theory
above a certain expressiveness (read as can express natural numbers with
addition and multiplication) can be consistent and complete. Assuming Peano
arithmetic is consistent, it can not be complete and complete means you can
prove all true facts expressible in the system within the system itself.

The (standard) proof of Goodstein's theorem uses ordinal numbers [2] which are
outside of Peano arithmetic and the Kirby–Paris theorem proves that there is
no proof inside Peano arithmetic [3].

[1]
[http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...](http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theorems)

[2] <http://en.wikipedia.org/wiki/Ordinal_number>

[3]
[http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goo...](http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goodstein.27s_theorem)

------
MojoJolo
Here's the research paper discussing and proves the conjecture.
<http://arxiv.org/pdf/1305.2897v1.pdf> It's a 130 page research paper
excluding references mostly containing proofs and formulas.

------
lquist
Can a mathematician explain the significance of this result please?

~~~
ColinWright
Briefly ...

To a large extent the primes appear to be distributed in a manner
indistinguishable from randomly. There seems underneath to be no reason to
believe that there are results like this. If they were random then there might
be numbers that cannot be expressed as the sum of 3 primes, but somehow
_every_ number ends up so expressible.

Why should that be true? Indeed, _is_ it true? This result says it is true,
but we really don't see why.

Understanding how the primes are distributed may have far-reaching
implications for cryptography, and possibly even for solving things like the
TSP. We just don't know, just as we initially never suspected that public-key
cryptosystems were possible, and would involved primes.

Random matrices is another area where a lot of research was done just because
people found it an interesting problem, and now there appear to be deep
connections with practical physics that may allow us to further bend the world
to our will.

Sometimes it's the chase that's exciting, never knowing what may turn out to
have world-changing implications.

~~~
gjm11
If I'm understanding correctly, you suggest that this is an example of the
prime numbers _not_ behaving randomly. I disagree.

Here's (more or less) the simplest possible random model of the primes. (It's
called the Cramer model.) Take P to be a random subset of the natural numbers
that includes each n>1 independently with probability 1/log(n). Then P is
kinda like the prime numbers, though it lacks some structure it should have
(e.g., it doesn't have the property that almost all elements are odd).

If you do this, then the expected number of ways to write a large n as the sum
of two elements of P is on the order of n/log(n)^2, which is far enough from 0
that I bet (though I haven't checked) that Pr(every number >= 4 is the sum of
two elements of P) is positive, and in fact probably quite big. This is kinda
analogous to the Goldbach conjecture; note that it isn't only about even n,
since in this model there can be lots of even "prime numbers".

I did a quick computer experiment, and it looks as if the probability is about
20%. In other words, if the primes were chosen at random according to the
Cramer model, then about 1/5 of the time the appropriately mangled Goldbach
conjecture would be true. (And the "odd Goldbach conjecture", which is what
Helfgott has just proved, would be true substantially more often than that.)

Now, of course the Cramer model is way too simple. In particular, it doesn't
know that almost all prime numbers are odd. So let's make it just a little
smarter by declaring that 2 is always in P, the other elements of P are always
odd, and that any odd n>1 is in P with probability 2/log(n). And now we should
switch back to the original form of the Goldbach conjecture, looking only at
even n. Well, according to my computer experiment the probability that the
Goldbach conjecture holds if we use P instead of the primes is about 96%!

[EDITED to add: and if we force P to get 2 and 3 "right" instead of just 2,
the probability goes up to about 99.5%. I fixed a few other things in this
edit too.]

In other words: _if the primes are basically random, then we should expect the
Goldbach conjecture to be true_.

~~~
robertk
Of course, your computer simulation completely disregards the possibility of
some very large (but rare) numbers (possibly of density zero) not being
expressible as such a sum. You merely verified it for low finite integers. I
am very unconvinced your percentages are accurate. In fact, they may be
completely wrong, as this issue is the entire crux of the Goldbach problem!

~~~
gjm11
My simulation disregards it, but _I_ don't. In this sort of random model, the
probability of any (specific) large number not being the sum of two "primes"
is extremely small (and decreasing rapidly as the number increases), which
means that the total probability of any (at all) large number not being the
sum of two "primes" is also extremely small.

In other words: in this model, with probability very close to 1 _all large
enough even numbers_ are sums of two "primes".

Let's put a little flesh on those bones. I'll stick with the unmodified Cramer
model for simplicity; the tweaked versions are fiddlier but not fundamentally
different. The probability that n=a+b is a "good" decomposition is
1/log(a)log(b); this is smallest, for fixed n, when a=b; so the probability
that any given decomposition "works" is at least 1/log(n/2)^2. There are n/2-2
of these, and they're all independent, so the probability that none of them
works is at most (1-1/log(n/2)^2)^(n/2-2), which is approximately exp(-n/(2
log(n/2)^2)). So the expected number of n beyond, say, 1000 for which that
happens is the sum of this for n from 1000 up. Unfortunately neither the sum
nor the obvious approximation as an integral has a closed form, but we can
compute it numerically; it's about 0.000279. That's the expected number of
failures above n=1000, and of course that's an upper bound on the probability
of at least one such failure.

So, the probability (in this model) of any Goldbach failures above n=1000 is
at most about 0.0003, and therefore the probability of any Goldbach failures
at all is at most that much bigger than the probability of any Goldbach
failures up to n=1000.

Note that although this is a statement with probabilities in, the
probabilities aren't _fractions of the integers for which Goldbach fails_.

The situation is analogous to the following. Suppose you flip a coin 10 times,
then 20 times, then 40 times, then 80 times, etc., and for each group of flips
you "win" if you get at least one head. Then, with probability about 99.9%,
you _always_ win: the probability of failure in successive groups drops so
fast that the contribution to the overall failure probability from all groups
other than the first is tiny.

Similarly, with the Cramer-model analogue of Goldbach's conjecture, the
probability that the conjecture fails "at n" decreases so rapidly with n that
almost all the probability of there being any failure at all comes from the
possibility of failures for small n.

And that's why you can get good estimates of the failure probability from
computer simulations that look only at finitely many n.

------
rdl
This sort of terrifies me. I know there's no known connection to factoring,
but still.

~~~
arethuza
Really? I thought it was fascinating and spent a few minutes running through
some examples... Quite a sensawunda moment.

~~~
rdl
My only real interest in number theory is applied number theory; i.e. the
basis of many schemes of asymmetric cryptography.

~~~
phillmv
Well… the weak Goldbach conjecture has been known for a couple hundred years
and my understanding is it's one of those things we've suspected was true but
just never found a solid way to prove that it was so.

This is a neat result, but no one's _surprised_ , per se.

~~~
jokoon
Being able to prove can also bring new pieces of mathematics.

~~~
rdl
I mostly just meant "any results in the space at all" might change things, and
really can only change things for the worse (if you have deployed
cryptosystems).

~~~
cantos
There are 2 mostly distinct branches of number theory additive and
multiplicative. Integer factorization is related to multiplicative number
theory, but not really additive (as far as is known). This is a result in
additive number theory so you shouldn't be any more concerned than say if a
major problem in graph theory or differential equations was solved.

The fact that the security of a cryptosystem can only degrade over time is
what I find so interesting about them. You can't just plug in a cryptosystem
and forget about it. If you want real security you will regularly have to make
changes.

------
ximeng
Also interesting work on twin primes this week:

[http://www.nature.com/news/first-proof-that-infinitely-
many-...](http://www.nature.com/news/first-proof-that-infinitely-many-prime-
numbers-come-in-pairs-1.12989)

~~~
mixmax
In my country social security numbers consist of two numbers, and mine
consists of two twin primes. Probably the only one in the country.

This is something I can only brag about on HN :-)

~~~
otibom
Finding mixmax's country of residence, social security number, name and
address is left as an exercise for the reader :)

~~~
mixmax
should definitely be doable - but using it to defraud me would be hard. Here
social security numbers are used as an identifier, you can't really get to
anything interesting with it. For that you'll need my passport, drivers
license, digital ID or something similar. You can probably go to the doctor
and claim to be me, but that's about it.

As a matter of fact a popular singer once published an album where the title
was his social security number to show that you can't do much with it.

------
MojoJolo
Curiosity. With this 130 page research paper, I'm really curious what's its
application in the near future. I'm sure there will be, I'm seeing this in the
branch of cryptography.

I'm imagining it to be implemented like a public-private key. Maybe the odd
number can be the private key. And three prime numbers can be the public keys.
The hash can't be decoded by just the single public key. It needs all three of
the public key to decode the hash. Well, with just abrupt thinking, I think it
can be more secure? (Not really sure about it though. Just a guess.)

~~~
lmm
I don't see such an application; given a number which is known to be a sum of
primes it's very easy to reverse this and extract the primes (unlike
multiplication, which is how RSA works). And for practical cryptography
processes it's always been perfectly reasonable to assume the goldbach
conjecture - even though there's no formal proof yet, it's true in such an
overwhelming majority of cases that you really wouldn't worry about hitting a
counterexample, the odds are far smaller than e.g. a hash collision.

~~~
Retric
If you add three large primes I don't see how you could quickly reverse that
process. May be possible that several set's of 3 primes also sum to that
number so you would need a process for finding all of them.

~~~
eterm
Factorisation is unique, the addition of 3 primes is not.

e.g. 29 can be written 5 + 11 + 13 or 3 + 3 + 23

So even if it were a difficult operation to reverse addition of 3 numbers, it
would be made easier by collisions.

~~~
pbhjpbhj
> _Factorisation is unique_ //

Um, no it isn't.

6, 4 are factors of 24; 2, 12 are factors of 24.

Probably you meant prime decomposition.

~~~
ramidarigaz
I think he means that prime factorization is unique (which is true).

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

~~~
pbhjpbhj
Isn't that what I said?

~~~
ramidarigaz
I could have sworn that last sentence wasn't there (my bad).

------
postit
Can't wait to use this to prove that are infinite primes. :P

~~~
SatvikBeri
Assume there are a finite number of primes. Let _p_ be the largest prime. Then
3 _p_ +2 is an odd number > 5, so it must be expressible as the sum of 3
primes. But since _p_ is the largest prime, the sum of 3 primes can never be
larger than 3 _p_ , contradiction.

------
sytelus
From Wikipedia:

 _Goldbach's conjecture is one of the oldest and best-known unsolved problems
in number theory and in all of mathematics._

If this proof is correct then it's a bigger deal than Fermat's Last Theorem.

~~~
lquist
This is a proof of Goldbach's weak conjecture
(<http://en.wikipedia.org/wiki/Odd_Goldbach_conjecture>), not Goldbach's
(strong) conjecture.

------
ajuc
So we now can take 2 greatest known prime numbers, substract them from some
HUGE odd number (bigger than (both these prime numbers) * 2 + 1), and we have
a new greatest known prime number?

Or am I missing sth?

EDIT: ah, I get it now, it's sum of SOME 3 prime numbers, but not neccesarily
these 2 that I've choosen to make it and another one, it may be 3 completely
different primes

~~~
azakai
Not every sum is of three primes.

15 = 5 + 5 + 5

which are three primes, but

15 = 2 + 3 + 10

which are not. The theorem says there there _is_ such a sum, not that all sums
are of that form.

------
asdr
Such papers MUST include source code available for further research
activities.

------
nicknash
Interesting that it involved a significant computational effort in verifying
the GRH up to a large finite constant.

~~~
igravious
And also that, among others, the four-colour theorem's only known proof is a
computerized check of hundreds of cases. Will this increasingly be the case?
Is this the way mathematics is going? Will any sufficiently difficult problem
from now on require a computational step? Does that mean that mathematicians
until now were using the non-computational subset of mathematics?

------
lifeformed
Why greater than 5? Don't 3 and 5 work as well?

3 = 1 + 1 + 1

5 = 3 + 1 + 1

Or do they mean different primes?

EDIT: Oops, I forgot about 1 not being prime.

~~~
yock
Not to be a "pile on", but Numberphile did a great video on why 1 is not
prime. <http://www.youtube.com/watch?v=IQofiPqhJ_s>

~~~
mratzloff
I understand "we got tired of saying 'excluding 1'", but I still don't grasp
how the product of zero primes is 1 and not 0.

~~~
jeffasinger
Multiplying 0 times is the "same" as raising something to the 0 power.

~~~
mratzloff
Oh, of course. x exp. 0 == 1. I got that he was using powers but my brain
didn't equate 2 exp. 0 with how he explained it for some reason.

------
wyck
At this point it's not about practical applications. I find the idea that
number's (number theory, primes, etc) could be Platonist very interesting,
meaning it's existence could be outside ourselves and not an archetype. In
other words, a mathematical structure doesn't describe a universe, it is a
universe.

Ya I know, It sounds very "hippy".

------
aidos
As someone who doesn't study this subject, intuitively it seems so unlikely. I
would guess that 130 pages of proves is probably a little too much to tackle
to try to get a handle on it :)

Very curious result though.

~~~
lmm
As I understand it this is largely a technical refinement of the argument from
Tao's earlier paper <http://arxiv.org/abs/1201.6656> , which is a bit shorter
and very readable by the standards of such papers (though admittedly not
exactly elementary)

------
chiaro
What are the other practical applications of number theory, primes and this
the results in the paper outside cryptography?

------
jart
Oh my god this is so much math

<http://arxiv.org/pdf/1305.2897v1.pdf>

------
erikb
It's a proof. That doesn't say it's true.

------
kokey
Will this be useful for PKI encryption?

------
wcfields
Well, duh, of course it's every odd number ___greater_ __than Five....

Because Seven eight Nine.

I'll be here all week folks. Try the fish.

------
Zenst
I thought this was already know, been using in a pet project for a few years,
though I don't use numbers near infinity and with that you have to give the
chap credit as it is due. Shame it is not a prime number of pages as well and
simpler in explanation.

Maybe for his next paper he could do:

"Every even number greater than five is the sum of three primes and one"
Though one is a funny prime and with that I did not say four primes too keep
the peace. Can reference his previous paper and with that get two papers for
less than 134 pages compared to only one proof for 133 pages.

No I can't say what my pet project is, not everybody does lottery tickets ;).

~~~
euccastro
Counterexample: 6.

~~~
Dylan16807
Okay off-by-one, greater than 7.

But more fun is 'every even number 10 and up is the sum of at most four
primes, at least one of them being 3.

~~~
Zenst
Hmm, this is also very very true, Curse you the Number TWO being a prime and
not poor Number ONE. The madness :0.

So maybe (5x2)+1 onwards, though wondering that any two primes +2 will alwys
mess that one up.

Still all that said nothing about the said even number not being also able to
be made up by three primes without adding +1 ;). But still Wondering now how
many clash's and with that it does get down to the case of:

    
    
      Any Two primes + 2 would be three prime numbers and also a even number, 
      we all know that too be true and fact, even if those primes are also the value of 2.
    
    

Numbers are such fun, more so when you question them, learn them and respect,
or indeed change them. They are still fun.

