## Extended Goldbach: every odd number the sum of 5 primes

Terence Tao has submitted a paper to arxiv: [1201.6656] Every odd number greater than 1 is the sum of at most five primes
Its abstract:
 We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood and Vinogradov, together with Vaughan's identity; our additional techniques, which may be of interest for other Goldbach-type problems, include the use of smoothed exponential sums and optimisation of the Vaughan identity parameters to save or reduce some logarithmic losses, the use of multiple scales following some ideas of Bourgain, and the use of Montgomery's uncertainty principle and the large sieve to improve the $L^2$ estimates on major arcs. Our argument relies on some previous numerical work, namely the verification of Richstein of the even Goldbach conjecture up to $4 \times 10^{14}$, and the verification of van de Lune and (independently) of Wedeniwski of the Riemann hypothesis up to height $3.29 \times 10^9$.
One can turn the Goldbach conjecture and similar problems into statements about certain integrals, but those integrals are VERY hard to do, and it has only been possible to find analytic bounds for those integrals for very large numbers. There are, in fact, 2 Goldbach conjectures
• Even: every even number x > 0 is the sum of at most 2 primes
• Odd: every odd number x > 1 is the sum of at most 3 primes
The even one implies the odd one, and they both imply the 5-prime one, but these implications do not work in reverse.

TT quotes:
• Chen and Wang for odd: x >= exp(exp(11.503)) ~ exp(99000) ~ 3.33*1043000
• Liu and Wang for odd: x >= exp(3100) ~ 2.1*101346
• Richstein for even: x <= 4*1014 ~ exp(33)
• Ramaré and Saouter for odd: x <= exp(28) ~ 1.14*1022 ~ exp(51) (their paper)
So there's a big gap between the analytic results and the numerical ones.

TT's paper is about closing that gap for the 5-prime case, and I will concede that I find it difficult to follow.

But if his reasoning is sound, then that suggests that we may not be far off from proving Goldbach's conjecture by a similar method, or else making it much easier to find a counterexample.

Could it be possible to find bounds from above for similar unsolved mathematical problems?

The Riemann hypothesis. It's about the zeros of the Riemann zeta function, the values that make it zero. Its trivial zeros are the negative even integers, and the rest are its nontrivial ones. The hypothesis states that all of them have real part 1/2.

Mersenne primes. Mersenne numbers 2prime-1 include primes as far as they have been searched: 243,112,609 – 1 is the largest currently known one. It is not known whether there is a finite or infinite number of them.

Fermat primes. Fermat numbers 22^n+1 include only 5 known primes: n = 0 to 4: 3, 5, 17, 257, 65537. For n = 5 to 32, the Fermat numbers are known to be composite, and it is not known whether or not there are any other Fermat primes.
 Go Terence! He's been on a roll lately. Can't comment on the paper, though; analytic number theory may as well be magic to me.