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Busy day in analytic number theory; Harald Helfgott has complemented his previous paper http://arxiv.org/abs/1205.5252 (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes.  (This improves upon a result of mine from last year http://terrytao.wordpress.com/2012/02/01/every-odd-integer-larger-than-1-is-the-sum-of-at-most-five-primes/ showing that such numbers are the sum of five or fewer primes, though at the cost of a significantly lengthier argument.) As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation).  [EDIT: the proof also relies on extensive numerical verifications of GRH that were performed by David Platt.]
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Sudarshan Parthasarathy's profile photoDiego Alonso Cortez's profile photoCharlie An's profile photoRicardo Blanco's profile photo
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Just to be clear: there is little to no hope that this technique could be extended to prove the Goldbach conjecture right? Because of the "parity problem"?
 
Why does the abstract in the sample that is displayed on G+ say that the ternary Goldbach conjecture asserts that every odd integer N is greater than 5? That's not right is it?
 
5 isn't the sum of exactly three primes.
 
Oh gotcha. That little odd word there changes things.
 It's still strange though because the conjecture is stated differently in the actual abstract.
 
There must exist a much simpler proof
 
What three prime factors divide 7?
 
+Tatyana Beketova, you are right. I'm a moron. I completely misread that. Thanks.
 
What a busy week for prime-number results! Just a few days ago it was the Twin-Prime conjecture that was improved radically.
 
then 3+2=5 because both are prime
 
3,2 are just two numbers not three
 
oh, see   i didnt read it all, only like, a few comments
 
+Vadim Lebedev : the conjecture holds true for odd numbers greater than 5 !

+Kaley Wood : 3+2=5 is true but we're talking about an odd number greater than 5 and can be represented as a sum of three prime numbers! in this case 3 and 2 is ok, but where is the third number !
 
If you read the conjecture it states for numbers > 5 (I know I'm repeating others)

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer N greater than 5 is the sum of three primes. 
 
you need three digits... 3+2 is only two.
 
Leona Machado, I'm not sure whether you've read what I wrote, the conjecture states for integers greater than 5, the first prime being 7.
 
Fascinating how quickly a discussion about a mathematical discovery dilutes into a pillow fight over who misread who.
 
Congratulations, I thought this had already been proven as using in a pet project for few years now.  Though not using numbers upto infinity, so I guess that may explain why.

You might also wish to do another paper and state:

"All even numbers over 5 are the sum of no more than four primes - see my previous paper and add +1 to each value"  One is a funny prime too many.  That would save a few tree's as well }->.
 
Now I can easily prove every odd number greater than 7 is the sum of four primes ^_^
 
+Eric Pouhier there is. proof follows:  all odd integers greater than 5 can be expressed as a sum of 3 primes. See Helfgott Theorem. QED. :)
 
All even numbers greater than 2 can be expressed as (2n + 1) + 3 where (2n+1) is an odd number ;).
 
+Thomas Egense
Can you please post the link to the new result on the twin prime conjecture that you referred to in your comment?
 
What I don’t understand is how one can check even using a supercomputer all odd numbers up to 10^30? Suppose one can check one trilion number per second. Then it would still take 10^21 seconds, or 3.17e+13 years…
 
That's awful! Three breakthrough of mathematics in this week from 2013-05-12 to 2013-05-18:
1. First proof that infinitely many prime numbers come in pairs.
2. Proof of the weak Goldbach Conjecture:Major arcs for Goldbach's theorem.
3. Prime abc Conjecture b== (a-1)/(2^c) put forward. http://oeis.org/A225759
 
As always, thank you so much, Dr Tao, for being the go-to person regarding a disproportionate amount of mathematical stuff!
 
congratulations  Harald Andrés Helfgott ........ nos has dado algo mas que estudiar en colegios y universidades ......... AQP-Perú
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Categorizing each positive integer by its divisibility type is very useful. Since 1 cannot be regarded unambiguously as either prime or composite, and allowing it to be both would violate the fundamental theorem of arithmetic, it is the lone member of a third (unnamed) divisibility type.
 
i have arived at the following proof and need your suggestions(sudarsh0706@gmail.com)whether it is right or not i cant believe it as the steps are too simple but still i thought posting it here ...
Let l =2q where q is any integer between 2 to infinity therefore l would be any even integer greater than 2.                  Let us assume that l is not equal to m+h where m and h are  odd prime  numbers
Let m = x+1 therefore here since m is odd x would be even
Similarly let h=y+1 where y would be even
m+h=(x+1)+(y+1)     =  x+y+2  . since here x  y and 2 are even integers therefore m+h would be even
thus our assumption would be wrong and 2q=m+h  i.e  2 (2……infinity)=m+h 
 
i have sum resultat of twin Goldbach conjecture if you have some time.
 
The limit is very likely to be 16. Here is why:

After doing quick study of the statistical distribution of prime gap numbers, it seems to me that the gaps are a geometric distribution, which makes a lot of sense. The probability of hitting a prime number seems to equal to 1/exp(exp(1)), or the inverse of e to the power of e. The inverse of the probability is 15.15426. Hence the smallest gap must be 16.

Regards,
Charlie
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