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Old 2007-04-22, 14:05   #1
robert44444uk
 
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Default A new very prime k

Working as I do with nice large k, I thought I would introduce you to k=531131527270075522241760982081252274580435

I don't know what the nash weight is here, because k is too large for the nash calculator I have.

This k looks to as if it might better the currently known ultimate in producing primes as it is a Payam 162 number (no k in the power series k*2^n-1 has a prime factor where the order base 2 of primes is 162 or less), and has many many more small primes than the average such k. n=19763 and 19764 provide the largest of 7 Sophie Germains, 6 of which are over n=100, and 13521 and 13523 produced a near miss.

I will soon complete testing to n=100,000. At the time of writing I have tested to 90,137 and it has produced 142 primes, well up there with the best of k's discovered in my + search, and because of the density of numbers requiring checking can be expected to produce many more up to 1,000,000.

The current record, plus or minus, is 169 primes for a k, on the plus side.
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Old 2007-04-24, 17:16   #2
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Quote:
Originally Posted by robert44444uk View Post

The current record, plus or minus, is 169 primes for a k, on the plus side.
Actually no, the record is 167 primes to n=300500 - can't add up

Progress, now with this big k is 143 primes and n=101317, and I only had 7 plus numbers better than this. Record for 143 primes...n=76635. all of these numbers had smaller nash weights.
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Old 2007-04-26, 13:24   #3
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Quote:
Originally Posted by robert44444uk View Post
I don't know what the nash weight is here, because k is too large for the nash calculator I have.
The Nash weight should be 9275
(That's quite close to the max. possible Nash weight of 10000).

A modified version of the Nash calculator (which now supports your ultra large values of k) is attached (source and Linux binary).
Attached Files
File Type: zip nash_big.zip (4.9 KB, 100 views)
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Old 2007-04-26, 13:41   #4
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A 42-digit k! And quite an attractive one. Shall we try team sieving? But reaching 1M is going to be a major task and will take us ages
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Old 2007-04-26, 13:49   #5
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Yeah, I am having troubles reaching n=700000 with my 9-digit k=736320585 with ~142 candidates in every 1000 range of n
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Old 2007-04-26, 15:11   #6
Thomas11
 
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Quote:
Originally Posted by Kosmaj View Post
A 42-digit k! ... Shall we try team sieving?
Note, that Geoff's srsieves are restricted to k<2^64 (or 2^63?). Only Phil's ksieve would be able to cope with a k like this.

Perhaps, one should consider a multiple k team sieve on a small set of about 10 k or so, which are within the capabilities of Geoff's srsieves (up to about 20 digits). The problem with those high weight ks is that you'll get lots of primes at the smaller ns, but you'll encounter large gaps once you reach the "interesting" regions of n>250000 or so (means: primes which would enter the Top5000 list). Taking a multiple k set would give some nice averaging over these gaps (as we already learned from our other team drives). But means even more LLR tests to reach n=1M, of course...

Last fiddled with by Thomas11 on 2007-04-26 at 15:12
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Old 2007-04-26, 16:37   #7
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Quote:
Originally Posted by Thomas11 View Post
Note, that Geoff's srsieves are restricted to k<2^64 (or 2^63?). Only Phil's ksieve would be able to cope with a k like this.
Actually I am already sieving as I work through the n, using good old NewPGen, which was adapted for payam numbers. You enter the factorisation of the n rather than the absolute value. At present I am up to 115 billion on the sieve from 100000 to 500000, and at the same time I am checking the earlier results of the sieve each day in pfgw, which I know is not the most efficient way, but at least I know what I am doing with pfgw.

If there is a way to split the Newpgen into segments for distributed computing, it might work.

In the meanwhile, the following are prime for n above 100000

101060
102124
106385

So now I have 145 primes, and have checked as far as 111185, and I have my second machine now, working on 130000 - 140000

Next week I will inherit two big desktops so I am hoping to start to generate some real work, instead of theorising and playing around with small primes.
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Old 2007-04-26, 16:46   #8
robert44444uk
 
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Quote:
Originally Posted by Thomas11 View Post
The Nash weight should be 9275
(That's quite close to the max. possible Nash weight of 10000).

A modified version of the Nash calculator (which now supports your ultra large values of k) is attached (source and Linux binary).
If you can create a windoze version I can check the 800 or so payam 162 numbers I have generated, and see if there is a larger Nash weight - either that or I can send you the file of numbers.
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Old 2007-04-26, 16:56   #9
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Originally Posted by Thomas11 View Post
The problem with those high weight ks is that you'll get lots of primes at the smaller ns, but you'll encounter large gaps once you reach the "interesting" regions of n>250000 or so
Yeah, but there is a new record to be had - 170 primes or more before n=300000, just think - the most prime k ever, record might stand for ages !!!!!!

BTW, the most prime k through the 100000 to 250000 range produced:

104186
105283
107911
122428
123760
125392
126478
129152
129850
137749
172113
174521
184521
202474
206409
214727
226714
231290
240057
243709

That is a lot of primes, and where they come up is anyones guess.

PS. this prime had a gap from 243709 to the recently discovered 300332. So in one way you are right. We could prove you wrong
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Old 2007-04-26, 17:28   #10
Thomas11
 
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Quote:
Originally Posted by robert44444uk View Post
Actually I am already sieving as I work through the n, using good old NewPGen, which was adapted for payam numbers. You enter the factorisation of the n rather than the absolute value. ...
Of course, I know about that feature. However, it is much slower than sieving a "non-factorized" k with NewPGen. I found Phil's ksieve runs much faster than NewPGen on k>2^31 (when k needs to be factorized in NewPGen).

A Windows binary of the Nash calculator is attached, though I don't know whether it works on a non-P4 (non-SSE2) machine...
Attached Files
File Type: zip nash_win_p4.zip (80.5 KB, 90 views)

Last fiddled with by Thomas11 on 2007-04-26 at 17:29
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Old 2007-04-29, 08:19   #11
Citrix
 
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Robert,

I am interested in finding k's that produce alot of primes. How did you come up with this k? Please explain your method.

Thanks!
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