Goldbach's weak conjecture

MattcAnderson · 2017-03-18, 10:51

Hi Mersenneforum,

The 3 primes problem, or weak Goldbach conjecture has been proved true by
Helfgott in 2013.

It simply states that all odd numbers < 7 are the sum of 3 odd prime numbers.

See
https://en.wikipedia.org/wiki/Goldba...eak_conjecture

In My Humble Opinion (IMHO) this was quite a proof.

wanted you to know.

Regards,
Matt

CRGreathouse · 2017-03-18, 23:32

Greater than 7, that is.

Yes, it was a big deal! The proof had three parts: major arcs, minor arcs, and verifying weak Goldbach on small values. The third part was joint work with Platt and was verified considerably further than necessary: to 8.875e30 when Helfgott needed only 10^27.

Here's my understanding of the major+minor arc decomposition. Instead of looking at the sum
$\sum_{p+q+r=N}1$
of the number of ways a given odd number N can be written as a sum of three primes, Helfgott looks at the related sum
$\sum_{p+q+r=N}\log p\log q\log r$
which is a weighted version of the first which is easier to work with analytically.

Next the sum is transformed into an integral
$\int_0^1 S(\alpha,x)^3 - \exp(-2\pi iN\alpha)d\alpha$
which is then split into two pieces: the major arcs, which are small intervals around rationals with small denominators, and the minor arcs which are everything else.

Actually, the integral Helfgott actually uses is a somewhat more complicated 'smoothed' version of this integral, but the basic idea is the same: it can be broken into major and minor arcs, and proving that it takes on positive values for a given N shows that weak Goldbach holds for that N.

Helfgott spends a paper describing how to bound the minor arcs very precisely, then has a somewhat easier task (because of the groundwork he laid) on the major arcs. Combining the two Helfgott proves that the smoothed integral is at least 0.000422 * N^2 when N is an odd number greater than 10^27, which together with the numerical verification with Platt proves the weak Goldbach conjecture.

Helfgott cites a lot of software used in the proof including PARI, Maxima, Gnuplot, VNODE-LP, PROFIL / BIAS, SAGE, and Platt’s interval-arithmetic package (based in part on Crlibm).

2017-03-18, 10:51	#1
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 5×11×13 Posts	Goldbach's weak conjecture Hi Mersenneforum, The 3 primes problem, or weak Goldbach conjecture has been proved true by Helfgott in 2013. It simply states that all odd numbers < 7 are the sum of 3 odd prime numbers. See https://en.wikipedia.org/wiki/Goldba...eak_conjecture In My Humble Opinion (IMHO) this was quite a proof. wanted you to know. Regards, Matt

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2017-03-18, 23:32	#2
CRGreathouse Aug 2006 3²×5×7×19 Posts	Greater than 7, that is. Yes, it was a big deal! The proof had three parts: major arcs, minor arcs, and verifying weak Goldbach on small values. The third part was joint work with Platt and was verified considerably further than necessary: to 8.875e30 when Helfgott needed only 10^27. Here's my understanding of the major+minor arc decomposition. Instead of looking at the sum $\sum_{p+q+r=N}1$ of the number of ways a given odd number N can be written as a sum of three primes, Helfgott looks at the related sum $\sum_{p+q+r=N}\log p\log q\log r$ which is a weighted version of the first which is easier to work with analytically. Next the sum is transformed into an integral $\int_0^1 S(\alpha,x)^3 - \exp(-2\pi iN\alpha)d\alpha$ which is then split into two pieces: the major arcs, which are small intervals around rationals with small denominators, and the minor arcs which are everything else. Actually, the integral Helfgott actually uses is a somewhat more complicated 'smoothed' version of this integral, but the basic idea is the same: it can be broken into major and minor arcs, and proving that it takes on positive values for a given N shows that weak Goldbach holds for that N. Helfgott spends a paper describing how to bound the minor arcs very precisely, then has a somewhat easier task (because of the groundwork he laid) on the major arcs. Combining the two Helfgott proves that the smoothed integral is at least 0.000422 * N^2 when N is an odd number greater than 10^27, which together with the numerical verification with Platt proves the weak Goldbach conjecture. Helfgott cites a lot of software used in the proof including PARI, Maxima, Gnuplot, VNODE-LP, PROFIL / BIAS, SAGE, and Platt’s interval-arithmetic package (based in part on Crlibm).