Topic: [ANN][GAP] Gapcoin - Prime Gap Search - New Math Algo - CPU / GPU - Zero Premine - page 62. (Read 287669 times)

Paбoтa

what are the news that such a course is raging?

News? IT'S HAPPENING!

if there are any news about GAP?

And here we go. Again.

The probable prime test on the endpoints is a quite small portion of time.  It's also mostly playing with semantics as the endpoints are those that failed the test we applied to the interior post-sieve candidates.  Most importantly, it also doesn't match the way the records have been stated over the last 20 years, and would lead to very different record results.  Specifically, the records are stored by gap size (with prime endpoints).

The test Dr. Nicely's cglp4 does is strong BPSW, and it's possible to add additional non-overlapping probable primality tests for little cost.

Proofs are possible.  On the 257 gapcoin record gaps, doing the verification including a primality proof of the endpoints took under 8 minutes on a single core of a home computer.  But the time grows rapidly as the size increases, and isn't feasible past 30k digits with today's software/hardware.  If you look at Nicely's results, gaps under 1000 digits typically have proven endpoints, which is quite a bit larger than gapcoin's results.

Note that item 1) could be left out entirely.

Item 2) by itself guarantees a gap.
In case 1) doesn't hold, the actual gap is simply bigger than claimed.

In fact for the sake of simplicity, 1) *should* be left out.

You end up with a simpler notion, consecutive composites, and with simpler
verification code, that no longer needs the uncertainty of probabilistic prime tests.

Finding first occurrence or record (first known occurrence) prime gaps tells us something about the structure of the primes.  How much is debatable.  The recreational/computational task of finding them may lead to new ideas.  It demonstrably leads to more efficient software, parts of which can be re-used for other tasks.

Finding new first occurrence gaps would be, in my opinion, more useful in terms of final results, as the results are concrete: "no prime under this point starts a longer gap" vs. "we did lots of work and this is the best we found".  The process involves doing an exhaustive search, and one problem would be verification that people actually *did* the work.  The final results are flawed if anyone cheats or if a block gets skipped.  How to do a quick verification isn't immediately obvious.  Lastly, the network could compute for years and find no actual result, but just move the "no results below x" forward.  Useful, but not exciting.

Finding "the next gap" is a much easier problem, and takes as long to verify as it does to find (very slightly less, but within a few percent).  Given the task of finding record prime gaps, improving the performance of finding the next gap (or primes in a range) is important.

Large prime gaps, like twin primes, are pointing out "hey, something interesting is happening here."  That is why it is more interesting than just listing billions of primes, or finding the next prime.  From a project point of view it has the nice features of "hard to find, easy to verify", relatively linear effort/reward (unlike factoring), and a bonus reward (we broke ## first known occurrence records) to keep up interest.

There are lots of computational problems like this, many with distributed projects.  I think record prime gaps is interesting and well structured for what gapcoin wants.  I think it's easier to see the possible mathematical benefits than, say, record Mersenne primes, large arithmetic progressions, or Riesel prime searches.

Yes, that should be all that is necessary.  To give you an idea, gapcoin has found 257 record gaps in the last 4 months across the whole network.  Verification of them all took 36.5 seconds on a single core.  There is still a very small chance the end points are pseudoprimes (this verification was with BPSW rather than a simple Fermat test).  After results get submitted to Dr. Nicely, proofs are eventually run on the endpoints if they're small enough, and that threshold is much higher than the sizes Gapcoin is producing.

It depends on "large" and "hard".  Gapcoin is finding primes in the 80-400 digit range, which isn't very large.  A verification for the high end is about half a second.  It gets more interesting at 4000 digits, taking maybe 10 minutes.  At 40,000 digits it's certainly possible but quite time consuming.  Martin Raab's monster 4.7 million digit prime can have the endpoints verified with good probable prime tests in 1-2 days but the interior looks like it would take months.

yes, except it uses a fast probabilistic primality test, which has a minuscule probability of being wrong.


These tests are very fast, and we only need a few thousand of them...

Does verifying them involve only

1) Make sure the two numbers are prime
2) Make sure there are no primes between them

And it seems like with large primes both those would be relatively hard

a lot of factual information, thanks guys

To the best of my knowledge, gapcoin's PoW's are doing effective work, and finding large prime gaps.  These are computationally difficult to find, but relatively easy to verify (very much so in the range gapcoin is looking at), and Gapcoin is producing results that some people care about.  Looking at the distribution might tell someone something, and particularly large merits might also provide more information (useful for bounds on c in the Cramér-Granville conjecture for instance).

In terms of efficiency, gapcoin is rapidly throwing darts with a blindfold on.  The CRT work at least points one in the right direction.  Other search methods are aiming the darts at high-probability areas which is why they generate more records.  There is arguably a benefit to searching in areas that wouldn't be covered by the traditional methods.  If gapcoin had more computational resources, it would generate more results of course, as would other methods.

Zhang's work, and the follow-on work from Tao et al. are theorems, so don't need any computational "proof of work", nor would any record prime gap result help prove, disprove, or provide supporting data for their work.  Record large prime gaps provide supporting data to hypotheses about prime gap upper bounds and averages.

Thanks! At least the PoW does something meaningful

As a nonmathematician trying to keep you two honest, it seems to me that all successful searches for prime gaps will be "proof of work" for any adequate theory that is not proved to the satisfaction of somebody.

More interesting might be the statement from http://www.newyorker.com/tech/elements/pi-prime-numbers-cybersecurity

"Zhang proved that no matter how far you go on the number line, even to the range where the numbers would fill many books, there will, on an infinite number of occasions, be two prime numbers within seventy million places of each other. (Other mathematicians have reduced this gap to two hundred and forty six.)"

If the number really is 246 then is the unique thing about "Gapcoin" that it searches for high merit primes? That it ignores the "246" that might not be as useful to theorists and looks for the "70million" so that others can do something or other on a slightly bigger field?

I believe this is just words strung together with no meaning.

Gapcoin is merely working "towards" the conjecture stated by Yitang Zhang.  The Rev2 setup is done by a member here built into the wallet, I don't think it was Jons work. 

It is simple really this is a large game of "Go".  Except there are no limits for players only the board is shortened and the Primes are allowed to build up to a very large  number along the "number line". 

70 million is the length of digits Yitang Zhang described as the bounded Gap for "Infinity" Twin Primes.  The other group is makes no logical arguments but is supported institutionally to build their own conjecture. 

Gapcoin digit "block" limit is set to 33.5 million in the Rev2 wallet I am unsure what it was in the first wallet I would have to download it and do "Getmininginfo" command. 

Gapcoin is the "Proof of work" for Yitang Zhang.  "Of course only time will tell"

Can you explain more why you think any gapcoin result proves Zhang's results (Bounded gaps between primes)?  It doesn't seem to follow at all.  Finding a large or small gap at some fixed point has no bearing whatsoever on a proof that for some even value k where k < 70,000,000, there exist infinitely many primes this distance apart.

What is "the Yitang Zhang algo"?

The algorithm description on github is a partial sieve, Fermat test for each remainder, then scan for large gaps.  This is not a novel technique and has nothing to do with Zhang.  Jonny's CTR work in GapMiner is doing a lot of startup work, but looks like it eventually comes down to sieve the range, Fermat test candidates, then scan for large gaps.  Nothing to do with Zhang.   Edit: When I say "not novel" I mean it's a standard method but there's lots of room for interesting implementation details, not that it isn't any good.

Gapcoin is currently the proof of work for Yitang Zhang conjecture which led to a split in the community those who "understood" it to be correct and those who attacked it. 

Terrence Tao has his own little group and gets much attention and awards but it seems up for debate to make your own choice.

The First Gapcoin with the password hash is not very efficient for mining using small blocks and sieve set like which Terrance Tao and others set as the minimum.  But it is not efficient for computers so it will never be solved.  They only deflect the attention away from Yitang Zhangs work.   

The Rev2 updated inwallet mining is the Yitang Zhang algo with a 33.5 million prime block with a 900,000 sieve set.  Which is half of Yitang Zhangs 70 million, but 33.5 million is the most efficient mining program.  There is no need to use the other programs unless you use a large set up and know how to use the programs themselves.  The inwallet mining for Rev2 is the highest rate it can be. 

Gapcoin=Yitang Zhang

But Gapcoin is the first to implement a variable reward system based on hashrate.  It is intelligent designed. 

Is the rev5.1 miner faster than the previous ones for mining or it is specific for larger shifts?