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.]
[1305.2897] Major arcs for Goldbach's theorem
Abstract: The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer N greater than 5 is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an ...
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- i have sum resultat of twin Goldbach conjecture if you have some time.Oct 22, 2013
- i'm in moroccoOct 22, 2013
- 2+2+3Jan 9, 2014
- 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.
CharlieJan 20, 2014
- Read thisJan 9, 2015
I have just completed a paper on additive number theory. One of the results is a proof to the strong GC. Would anyone be willing to review and provide feedback?