20170318, 10:51  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{3}·149 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 
20170318, 23:32  #2 
Aug 2006
5987_{10} 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, VNODELP, PROFIL / BIAS, SAGE, and Platt’s intervalarithmetic package (based in part on Crlibm). 
20170319, 18:23  #3 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{3}×149 Posts 
Goldbach's weak conjecture
Hi Mersenneforum,
In 2013, Harald Helfgott proved Goldbach's weak conjecture. See  Wikipedia Article about Goldbach's weak conjecture I wrote some Maple (Maple is a computer algebra system.) code in order to shine a light on Harald's accomplishment. Regards, Matt 
20170319, 19:01  #4 
"Robert Gerbicz"
Oct 2005
Hungary
3113_{8} Posts 

20170319, 20:18  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{3}×149 Posts 
Hi again Mersenneforum,
So I was unable to improve my computer code significantly. I found better code at Online Encyclopedia of Integer Sequences My code cannot calculate gwc(7000) in under 102 seconds. At least it produces correct results. (as far as I tested it.) Regards, Matt 
20170319, 22:24  #6 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
3^{2}×101 Posts 
I added a simple Perl program based on Luschny's algorithm. It takes ~5 seconds on my laptop for a(7000).
Code:
perl E 'use ntheory ":all"; sub a007963 { my($n,$c)=(shift,0); forpart { $c++ if vecall { is_prime($_) } @_; } $n,{n=>3,amin=>3}; $c; } say "$_ ",a007963(2*$_+1) for 0..100;' For OEIS use it's almost getting to the point where it would be worthwhile to add a isprime option to forpart as seems to keep coming up as a restriction. It would make this function trivial. 
20170319, 22:36  #7 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
909_{10} Posts 
It was simple (5 lines) to add a prime restriction to my forpart, and it runs this function about 20x faster, since forpart's C code is doing all the work. 0.25s for a(7000).

20170320, 03:59  #8 
Aug 2006
1011101100011_{2} Posts 
Here's a translation of the original program into GP:
Code:
gold(n)=sum(a=2,n\3,sum(b=2,a,sum(c=2,b,prime(a)+prime(b)+prime(c)==n))) Code:
gold1(n)=my(s,t); forprime(p=(n+2)\3,n6, t=np; forprime(q=t\2,min(t3,p), if(isprime(tq), s++))); s These should definitely be improvable  sieving should replace the final primality test for fairly large gains. But not bad for a oneline script. 
20170320, 04:18  #9 
Aug 2006
13543_{8} Posts 
13 minutes, 52 seconds to get a(10^6) = 235945824.
While I'm here I'll also advertise my post on the other Goldbach thread Matt started recently, where I attempt to summarize Helfgott's basic argument. Last fiddled with by CRGreathouse on 20170320 at 04:20 
20170320, 04:19  #10 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
3^{2}·101 Posts 
The OEIS sequence was fun, but to be clear it is horrifically slow in terms of the simpler question for the ternary Goldbach conjecture.
Helfgott in his intro describes the covering for all n < 10^27 "really a minor computational task." 1.2.2 discusses some methods. Using Oliveira e Silva et al.'s results of the binary Goldbach conjecture to 4*10^18 make it quite reasonable  this rather underplays the considerable effort of that check. The paper coauthored with Platt points out how useful that result was. He notes it took 25 hours on a single core to finish the verification. I wrote a simple checker and it's quite fast at first. Helfgott and Platt use a method that saves time on the primality proof by spending more time on the selection. I just used a simple prec_prime + is_provable_prime loop. The precprimes themselves are less than 2 hours on a single core to reach 10^27. The proofs are very fast until 3*10^24, then start sucking up quite a bit of time. I'm not sure it'd beat 25 hours, but I don't think it'd be too far off. 
20170320, 04:35  #11  
Aug 2006
5,987 Posts 
Quote:
Quote:


Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Goldbach Conjecture  MattcAnderson  MattcAnderson  4  20210404 19:21 
Factorial and Goldbach conjecture.  MisterBitcoin  MisterBitcoin  17  20180129 00:50 
Goldbach's Conjecture  Patrick123  Miscellaneous Math  242  20110315 14:28 
Goldbach's conjecture  Citrix  Puzzles  3  20050909 13:58 