mersenneforum.org Goldbachs Conjecture
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2017-01-09, 16:15   #1
MisterBitcoin

"Nuri, the dragon :P"
Jul 2016
Good old Germany

35F16 Posts
Goldbachs Conjecture

I love this conjecture. A lot of time I have wasted...

I´d like to restart research on it. And I could need some help. ;)

What is Goldbachs conjecture?

Goldbach’s Conjecture says that every even number larger than four is the sum of two odd prime numbers, for example 6 = 3+3, 8 = 3+5, 60 = 43+17, and so on.
Christian Goldbach first proposed this in 1742, in a letter to Swiss mathematician Leonhard Euler and it sounds easy enough to understand. Unfortunately, it’s not easy to prove and Goldbach’s Conjecture remains firmly on the ‘to do’ list of mathematical challenges to crack.

All important research was done by Tomás Oliveira e Silva, he hitted the value 4x10^18.
Why not going a bit deeper?

http://sweet.ua.pt/tos/goldbach.html
https://www.egi.eu/use-cases/researc...hs-conjecture/
And a dissertation as Attachment.

But we need to write an own programm. I´d asked him about porting it into BOINC.

"Nope. My program was partially written in assembly and as a limit of
(30*2^26)^2, which is slightly larger than 4*10^18. So I already did
all that could be done with the program. Note that I do not distribute
source code and I do not use primality tests. I use a very efficient
segmented Eratosthenes sieve, which is something that can be used,
perhaps, up to 10^22 (useless near 10^50, for example)."
Attached Files
 mcom2787.pdf (487.3 KB, 358 views)

 2017-01-09, 16:20 #2 rogue     "Mark" Apr 2003 Between here and the 3·17·131 Posts Such a program could only disprove the conjecture, by finding an even number that cannot be written as the sum of two primes. If a program cannot find an exception, it doesn't prove anything. IMO running such a program to be a waste of time unless one has some ideas on how to find an exception. Last fiddled with by rogue on 2017-01-09 at 16:24
2017-01-09, 16:37   #3
MisterBitcoin

"Nuri, the dragon :P"
Jul 2016
Good old Germany

35F16 Posts

Quote:
 Originally Posted by rogue Such a program could only disprove the conjecture, by finding an even number that cannot be written as the sum of two primes. If a program cannot find an exception, it doesn't prove anything. IMO running such a program to be a waste of time unless one has some ideas on how to find an exception.
One problem (i think it is a problem) is: At which point can we say the conjecture is proven?
I´d like to use the amount of all particels in the universe. (close to 10^80) Source:https://www.quora.com/How-many-parti...n-the-universe

It might be time wasting, but disaproving it will be a massive shock.

2017-01-09, 16:54   #4
CRGreathouse

Aug 2006

3×1,993 Posts

Quote:
 Originally Posted by rogue IMO running such a program to be a waste of time unless one has some ideas on how to find an exception.
Personally, I find a lot of value in knowing the maximal prime gaps, which fell out of TOS's Goldbach work. So at the least I don't think it's a waste of time. (On the other hand doing the search inefficiently would be a waste.)

 2017-01-09, 16:55 #5 MisterBitcoin     "Nuri, the dragon :P" Jul 2016 Good old Germany 11010111112 Posts Fast implementation of the segmented sieve of Eratosthenes Might be interessting: http://sweet.ua.pt/tos/software/prime_sieve.html
 2017-01-09, 17:31 #6 science_man_88     "Forget I exist" Jul 2009 Dumbassville 2·5·839 Posts using the equidistant from any number version we haveif the composite is of form 0 mod 6 no distances that land on 1 or 5 mod 6 can be eliminated easily( at least not without switching to a higher mod) if the composite is of from 1 mod 6 possible distances include all numbers 0 mod 6 but for say a combinations of 1 and 5 mod 6 or a combination of 5 and 5 mod 6 the first possible distances grow because if we call the composite n, n-2 is 5 mod 6 but n+2 isn't either that could be prime. if n is 2 mod 6 now we have 5 mod 6 three away on either side so distances that are 3 mod 6 are allowed. no primes of form 1 mod 6 work. if n is 3 mod 6 1 mod 6 and 5 mod 6 are equidistant either side. no primes can be eliminated as possible. if n is 4 mod 6 is equidistant from primes of form 1 mod 6. if n is 5 mod 6 it's equidistant from primes of form 5 mod 6 I've been told to post this through PM.
2017-01-09, 17:43   #7
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

2·2,689 Posts

Quote:
 Originally Posted by MisterBitcoin One problem (i think it is a problem) is: At which point can we say the conjecture is proven? I´d like to use the amount of all particels in the universe. (close to 10^80) Source:https://www.quora.com/How-many-parti...n-the-universe It might be time wasting, but disaproving it will be a massive shock.
You can say it's proven when you.... have a proof. Demonstrating values that work are unrelated to a proof, no matter how high the value you demonstrate.
For a better-known example, compare prp tests to primality proofs for very large numbers. PRP tests for large numbers are much much better than 1 in 10^85 chance of composite, yet those are not considered proof.

2017-01-09, 17:54   #8
MisterBitcoin

"Nuri, the dragon :P"
Jul 2016
Good old Germany

863 Posts

Quote:
 Originally Posted by VBCurtis You can say it's proven when you.... have a proof. Demonstrating values that work are unrelated to a proof, no matter how high the value you demonstrate. For a better-known example, compare prp tests to primality proofs for very large numbers. PRP tests for large numbers are much much better than 1 in 10^85 chance of composite, yet those are not considered proof.
You´re right. And, after some time of deep mind working, I cant get a good reason to start working as a private person.
There a lot of unsolve able problems. Even TOS was using resources from the NICS. The amount of CPU-time for douple-checking 4x10^17 up to 5x10^17 will be quite a few years, one a single core. And you´ll need a lot of disk space, too.
Overall, it´s not making sense to start in it as a private person.

If his project will be re-activted he´ll let me know this. He states this in his reply.
Last words for now: Thanks all for replying.

2017-01-09, 19:14   #9
CRGreathouse

Aug 2006

135338 Posts

Quote:
 Originally Posted by MisterBitcoin The amount of CPU-time for douple-checking 4x10^17 up to 5x10^17 will be quite a few years, one a single core. And you´ll need a lot of disk space, too.
Why a lot of disk space? I would expect this to use virtually no disk space at all.

2017-01-09, 20:39   #10
MisterBitcoin

"Nuri, the dragon :P"
Jul 2016
Good old Germany

863 Posts

Quote:
 Originally Posted by CRGreathouse Why a lot of disk space? I would expect this to use virtually no disk space at all.
I´ll do some kind of research on it, only if TOS would start aswell. On my system I have 16GB RAM (I can douple the amount, but money is missing).
I´m not sure if this amount (lets say 12GB) is enough of run the full range.

But, first off all: I´m running out of time, too. Only 1 week, then I´m going on assembly. RWE is calling me for 4 weeks. A quit dangerous job, the heaters have still about 40-50°C.
10 hrs/day. (No weekend work YAY)
I don´t think I´ll do anythink after work. (only shower and bed)

 2017-01-09, 21:07 #11 danaj   "Dana Jacobsen" Feb 2011 Bangkok, TH 90910 Posts Like Charles, I think there are some good things to come out of the process. Mostly improving software, much as how finding large Mersenne primes is a waste of time in many ways, but yields benefits of faster software and better underlying operations. This benefits other code that I care about. With Oliveira e Silva's work, we got a cache friendly sieve implementation. primesieve uses a version of it at large sizes. It's not very memory friendly however. We also got some maximal prime gaps and know that all under 4e18 were maximal rather than "largest known". Somewhat similar, people have spent lots of computer time computing massive numbers of zeros of the Riemann Zeta function. At this point I doubt they're holding their breath waiting for a falsifying result. Büthe (2014) gives a nice result given those results, which is useful for practical computations. Forget 10^80, just getting to 2^64 would be a *massive* undertaking. And prove nothing other than be a nifty little result that it holds for all 64-bit numbers, which might be important to some implementation of something or other. But you still need an actual proof, as Curtis notes. I like his example of primality -- BPSW has been tested (through non-trivial means) for all 64-bit numbers, and various projects have attempted to target counterexamples to weakened tests to no avail. No matter how many trillions upon trillions of numbers we test, it's still not a proof for >64-bit numbers. I used primesieve on a 20-core machine (dual xeon workstation) to do an exhaustive check of prime gaps. It was 2 weeks to go from 4.000e18 to 4.001e18. Memory use would be pretty obnoxious for BOINC users. I saved largish gaps, but didn't do any Goldbach conjecture work, and didn't store counts at intervals. If a project did start, I think it'd be important to note what, for each interval, should be checked and what statistics to store. Disk space shouldn't really be an issue -- you're just storing something like the start point and count for a known interval length (e.g. the 2^30 intervals in the paper you give). Note running 20-way is going to result in under 20x speedup. I didn't try comparing different parallelization strategies. On a i7-6700K it looks like 4-core gives a 2.5x speedup with parallel primesieve.

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