2017-03-18, 10:51 | #1 |
"Matthew Anderson"
Dec 2010
Oregon, USA
3×13×17 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 |
2017-03-18, 23:32 | #2 |
Aug 2006
2^{2}×1,493 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 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 which is a weighted version of the first which is easier to work with analytically. Next the sum is transformed into an integral 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). |
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