List of most small twins of form k*2^n+/-1 - Page 19

roger · 2010-11-29, 06:59

and I guess I'll continue gdbarnes' work of k<1M - from what I gather he finished at n=48000, so I'll go from there - to 50,000 for now.

roger · 2010-12-04, 08:09

Update:
found 122 twins 10100<n<10635 1M-10M given that gdbarnes has searched the k's<1M.

no twins so far from 48000-48217 under 1M (by the way gdbarnes: where exactly did you leave off in this range?)

We've now found 320 twins above n=10000, with only 3 of these above n=50000.

I've also done some benchmarks on my computer (2.9GHz) for a variety of n-values:

Code:

		time (seconds)
n	k~1000	k~15M	k~125M	k~1.5G
10000	0.10	0.10	0.13	0.13
33200	1.09	1.30	1.32	1.36
34528	1.64	1.96	1.67	1.71
48001	2.27	3.37	3.26	3.51
50363	2.37	3.70	3.41	3.75
70727	4.68	7.20	7.66	7.42
75937	4.50	7.70	7.82	7.82
83047	6.57	9.50	10.21	9.47
98689	9.97	15.40	15.46	15.65
123456	12.30	23.40	23.68	23.56
145235	22.87	36.17	33.34	33.14
245235	69.29	103.70	96.32	94.92
345235	151.32	271.96	253.27	256.96
545235	390.57	601.51		
745235	776.94	1393.70		
945235	1276.95	2020.52		
1245235	2458.07	3429.71

Trendline for k's around 1000 is 4e-10*n^2.0946
for k's around 15million is 2e-10*n^2.173
Average increase of test time between k=1000 and 15M is 1.498 but is 1.607 for n's<50000.
For higher k-values (15M, 1.5G) the increase in test time from k~1000 is about 1.59x above n=40000.
Average trendline for k~1M+ is 3e-10*n^2.1579

I also doublechecked the two smallest k-value twins on the top-20 list, and both are the lowest-k twin values:

108615*2¹¹⁰³⁸²+-1 is prime
598899*2¹¹⁸⁹⁸⁷+-1 is prime

EDIT: also, Gary, is the n=100,000 twin you found the lowest k for that n?
same question for n=195000 and n=333333 from TPS

Vato · 2010-12-04, 13:40

For your last question, see http://www.mersenneforum.org/showthread.php?t=13667
combined with the fact it was an exhaustive search.

kar_bon · 2010-12-04, 17:34

Quote:

Originally Posted by Vato

For your last question, see http://www.mersenneforum.org/showthread.php?t=13667
combined with the fact it was an exhaustive search.

I think, roger meant if the found k-values for the twin primes for n=195000 and n=333333 are the lowest ones.

The post you cite was the test of all n-values of those 2 k-values and found primes.

Vato · 2010-12-05, 03:30

True, re that post, but it was an exhaustive search, and all k below those values were tested afaik (at least for n=333333). Raw data available at http://www.primegrid.com/pubresults.php or failing that ask John.

roger · 2010-12-06, 03:04

Now doublechecking n=169690 - confirmation should take about 11 days.

As for the current ranges:
10100<n<11000 pass one (k<1M): 150 found, 79% done
10100<n<11000 pass two (1M<k<10M): 20 found, 10% done
10100<n<11000 total: 192 found; 18.9% done
48000<n<50000 pass two: 0 found, 14.8% done (suspended while doublechecking occurs).

146878935*2⁵⁰³⁶⁴+-1 is prime! (15161 digits)

The links posted re n=195000 and n=333333 are unfortunately incomplete, covering only a portion of the search.

Can anyone check with those involved what k-value the testing started at?

Thanks!

Also: I compiled a list of our results, to see the distribution of the k-values:
eg there are 82 lowest twin k-values under 1000 for n<1000.

Code:

(k-value)
<1e3	1e3-4	1e4-5	1e5-6	1e6-7	1e7-8	1e8-9
82	166	430	311	10	0	0	n<1000
2	11	115	559	313	0	0	1000-2000
0	4	35	294	641	26	0	2000-3000
0	6	20	183	677	114	0	3000-4000
0	1	5	118	624	252	0	4000-5000
0	1	6	79	499	415	0	5000-6000
0	1	3	57	415	523	1	6000-7000
0	0	5	51	337	601	6	7000-8000
1	1	3	30	283	661	21	8000-9000
0	0	2	30	236	670	62	9000-10000
1	2	19	171	159	80	12	n>10000 (so far)

ellipse · 2010-12-07, 04:12

Quote:

Originally Posted by roger

The links posted re n=195000 and n=333333 are unfortunately incomplete, covering only a portion of the search.

Can anyone check with those involved what k-value the testing started at?

Thanks!

n=195000 has been searched from k=1:

http://www.mersenneforum.org/showthread.php?t=5775

I don't know what k value n=333333 started at.

roger · 2010-12-07, 12:34

n=195000 added to the chart.

10100<n<11000 1e6<k<1e7 94% done
10100<n<11000 1e7<k<2.5e7 22.4% done
total found so far: 244

48000<n k<1e6 14% done
switching to using 0<k<5e6 at n=48500.

n=169690 is currently at 5.342M (16% of the found twin value).

Also, trendline has been optimized for twin-k approximation:
(0.1909*n^2.0744)*(2.7266*n^-0.115)

for example, this gives an expected twin-k value of 12.070G for n=195000 with a found value of 2.003G (16.6% of expected value).

This was done by calculating the averages of the differences of 100 twin k-values at a time, creating a trendline. This was then added to the previous calculation of the theoretical twin-k value (above).

I've tried to make a weighted score formula, maybe to get a bit more interest with some friendly competition.
$\frac{\sqrt{k} * n^{1.5} * time)}{1e9}$

This attempts to give n a non-linear weight, as well as incorporate time taken. If anyone has suggestions for this, please post them!

Example scores:
n=10100 k=1e6 = 0.182
n=10100 k=1e9 = 5.756
n=34528 k=1e6 = 6.483
n=34528 k=1e9 = 205.000
n=50000 k=1e6 = 37.977
n=50000 k=1e9 = 1200.935
n=195000 k=2G = 243987.756
etc

Happy hunting!

roger · 2010-12-12, 21:42

10100<n<11000 is now over half done:
10100<n<11000 0<k<1e7 100% done
10100<n<11000 1e7<k<2.5e7 69% done
10100<n<11000 2.5e7<k<5e7 43% done
total found so far: 459

n>10000 k<1e6 has now reached n=48385 (switching to a range of 0-5M at n=48500).

Other ranges as follows:

Code:

n min   n max   k-range done to n=
10100	11000	0-1M	11000
10100	11000	1-10M	11000
10100	11000	10-25M	10725
10100	11000	25-50M	10486
10000	48500	0-1M	48385
40000	40013	1-5M	40013
42000	42002	1-10M	42002
42003	42005	1-5M	42005
45058	45067	1-5M	45067
45068	45070	1-10M	45070
45071	45075	1-5M	45075
48000	48500	1-5M	48000
48500	50000	0-5M	48500
48500	50000	5-10M	48500
48000	50000	10-25M	48000
50000	50005	0-1M	50005
55056	55071	0-5M	55071
76067	76100	0-2M	76100
76101	76130	0-1M	76130
80000	80010	0-1M	80010
123456	123476	0-10M	123467

Exceptions to above (working on filling in gaps):
n=40001 done to 5.75M
n=40002 done to 20M
n=40003 done to 10.68M
n=40004 done to 50M
n=50005 done to 8.98M
n=76073 done to 1M (sieved, testing soon)
n=76074 done to 0.191M (sieved, testing soon)
n=76084 done to 8.589M
n=76092 done to 1M (sieved, testing soon)
n=76096 done to 0.504M (sieved, testing soon)
n=76097 done to 5M
n=123457 done to 4.376 (currently testing)

I know it's really fragmentary; I'll try to shore it up a bit while my other cores are working on the 10100<n<11000 range.

Current ranges time to finish:
10100<n<11000 1e7<k<2.5e7 20.2 hours
10100<n<11000 2.5e7<k<5e7 90.4 hours
n=123457 k<1e7 3.3 hours
n=169690 doublecheck 72.5 hours

I'm working on a chart for slightly better k-ranges and pmax values given the higher n-values we've arrived at:

Code:

	k range (1e6)	n-value (pmax in 1e9)
Pass	initial	final	10000	20000	30000	40000	50000
1	0	5	N/A	N/A	N/A	N/A	110
1.5	1	5	10	25	50	90	
2	5	10	20	30	60		
3	10	25	40	75	150		
4	25	50	50	130	225		
5	50	100	70				
6	100	200	100				
7	200	300	100				
8	300	400	100				
9	400	500	100

roger · 2010-12-16, 21:42

33218925*2¹⁶⁹⁶⁹⁰+-1 confirmed as lowest twin.

10100<n<11000 k<2.5e7 done.
10100<n<11000 2.5e7<k<5e7 73% done
10100<n<11000 5e7<k<1e8 starting
total found: 593 = 65.9% done.

10000<n<48500 k<1e6 done.
11000<n<48500 1e6<k<5e6 starting

All lower k-searched n's within ranges posted earlier are done; those with k-ranges above the range remain as I'm not bringing the entire range up to whatever value for a single excessive k, at least not yet.

Not really much to add today; hopefully one of the higher-n twins will pop up soon!

roger · 2010-12-29, 01:52

Alright, time for some updating after Christmas holidays:

10100<n<11000 k<2.5e7 done.
10100<n<11000 2.5e7<k<5e7 97.8% done [16 hours remaining]
10100<n<11000 5e7<k<1e8 72.6& done. [211 hours remaining]
total found: 754.

10000<n<48500 k<1e6 done.
total found: 186.
11000<n<48500 1e6<k<5e6 5% done (at n=12890)
total found: 141.

123456<n<123476 k<1e7 85% done (74% of n=123474 done)

Nothing major to report unfortunately, just a status update.

2010-11-29, 06:59	#199
roger Oct 2006 2²×5×13 Posts	and I guess I'll continue gdbarnes' work of k<1M - from what I gather he finished at n=48000, so I'll go from there - to 50,000 for now. Last fiddled with by roger on 2010-11-29 at 07:08

2010-12-04, 08:09	#200
roger Oct 2006 2²·5·13 Posts	Update: found 122 twins 10100<n<10635 1M-10M given that gdbarnes has searched the k's<1M. no twins so far from 48000-48217 under 1M (by the way gdbarnes: where exactly did you leave off in this range?) We've now found 320 twins above n=10000, with only 3 of these above n=50000. I've also done some benchmarks on my computer (2.9GHz) for a variety of n-values: Code: time (seconds) n k~1000 k~15M k~125M k~1.5G 10000 0.10 0.10 0.13 0.13 33200 1.09 1.30 1.32 1.36 34528 1.64 1.96 1.67 1.71 48001 2.27 3.37 3.26 3.51 50363 2.37 3.70 3.41 3.75 70727 4.68 7.20 7.66 7.42 75937 4.50 7.70 7.82 7.82 83047 6.57 9.50 10.21 9.47 98689 9.97 15.40 15.46 15.65 123456 12.30 23.40 23.68 23.56 145235 22.87 36.17 33.34 33.14 245235 69.29 103.70 96.32 94.92 345235 151.32 271.96 253.27 256.96 545235 390.57 601.51 745235 776.94 1393.70 945235 1276.95 2020.52 1245235 2458.07 3429.71 Trendline for k's around 1000 is 4e-10n^2.0946 for k's around 15million is 2e-10n^2.173 Average increase of test time between k=1000 and 15M is 1.498 but is 1.607 for n's<50000. For higher k-values (15M, 1.5G) the increase in test time from k~1000 is about 1.59x above n=40000. Average trendline for k~1M+ is 3e-10n^2.1579 I also doublechecked the two smallest k-value twins on the top-20 list, and both are the lowest-k twin values: 1086152¹¹⁰³⁸²+-1 is prime 5988992¹¹⁸⁹⁸⁷+-1 is prime EDIT: also, Gary, is the n=100,000 twin you found the lowest k for that n? same question for n=195000 and n=333333 from TPS Last fiddled with by roger on 2010-12-04 at 08:18*

2010-12-04, 13:40	#201
Vato Jan 2009 24₈ Posts	For your last question, see http://www.mersenneforum.org/showthread.php?t=13667 combined with the fact it was an exhaustive search. Last fiddled with by Vato on 2010-12-04 at 13:41

2010-12-06, 03:04	#204
roger Oct 2006 104₁₆ Posts	Now doublechecking n=169690 - confirmation should take about 11 days. As for the current ranges: 10100<n<11000 pass one (k<1M): 150 found, 79% done 10100<n<11000 pass two (1M<k<10M): 20 found, 10% done 10100<n<11000 total: 192 found; 18.9% done 48000<n<50000 pass two: 0 found, 14.8% done (suspended while doublechecking occurs). 1468789352⁵⁰³⁶⁴+-1 is prime! (15161 digits) The links posted re n=195000 and n=333333 are unfortunately incomplete, covering only a portion of the search. Can anyone check with those involved what k-value the testing started at? Thanks! Also: I compiled a list of our results, to see the distribution of the k-values: eg there are 82 lowest twin k-values under 1000 for n<1000. Code: (k-value) <1e3 1e3-4 1e4-5 1e5-6 1e6-7 1e7-8 1e8-9 82 166 430 311 10 0 0 n<1000 2 11 115 559 313 0 0 1000-2000 0 4 35 294 641 26 0 2000-3000 0 6 20 183 677 114 0 3000-4000 0 1 5 118 624 252 0 4000-5000 0 1 6 79 499 415 0 5000-6000 0 1 3 57 415 523 1 6000-7000 0 0 5 51 337 601 6 7000-8000 1 1 3 30 283 661 21 8000-9000 0 0 2 30 236 670 62 9000-10000 1 2 19 171 159 80 12 n>10000 (so far) Last fiddled with by roger on 2010-12-06 at 03:27*

2010-12-16, 21:42	#208
roger Oct 2006 2²×5×13 Posts	332189252¹⁶⁹⁶⁹⁰+-1 confirmed as lowest twin. 10100<n<11000 k<2.5e7 done. 10100<n<11000 2.5e7<k<5e7 73% done 10100<n<11000 5e7<k<1e8 starting total found: 593 = 65.9% done. 10000<n<48500 k<1e6 done. 11000<n<48500 1e6<k<5e6 starting All lower k-searched n's within ranges posted earlier are done; those with k-ranges above the range remain as I'm not bringing the entire range up to whatever value for a single excessive k, at least not yet. Not really much to add today; hopefully one of the higher-n twins will pop up soon! Last fiddled with by roger on 2010-12-16 at 21:43*

2010-12-05, 03:30	#203
Vato Jan 2009 2²·5 Posts	True, re that post, but it was an exhaustive search, and all k below those values were tested afaik (at least for n=333333). Raw data available at http://www.primegrid.com/pubresults.php or failing that ask John.

2010-12-07, 12:34	#206
roger Oct 2006 2²·5·13 Posts	n=195000 added to the chart. 10100<n<11000 1e6<k<1e7 94% done 10100<n<11000 1e7<k<2.5e7 22.4% done total found so far: 244 48000<n k<1e6 14% done switching to using 0<k<5e6 at n=48500. n=169690 is currently at 5.342M (16% of the found twin value). Also, trendline has been optimized for twin-k approximation: (0.1909n^2.0744)(2.7266n^-0.115) for example, this gives an expected twin-k value of 12.070G for n=195000 with a found value of 2.003G (16.6% of expected value). This was done by calculating the averages of the differences of 100 twin k-values at a time, creating a trendline. This was then added to the previous calculation of the theoretical twin-k value (above). I've tried to make a weighted score formula, maybe to get a bit more interest with some friendly competition. $\frac{\sqrt{k} n^{1.5} * time)}{1e9}$ This attempts to give n a non-linear weight, as well as incorporate time taken. If anyone has suggestions for this, please post them! Example scores: n=10100 k=1e6 = 0.182 n=10100 k=1e9 = 5.756 n=34528 k=1e6 = 6.483 n=34528 k=1e9 = 205.000 n=50000 k=1e6 = 37.977 n=50000 k=1e9 = 1200.935 n=195000 k=2G = 243987.756 etc Happy hunting!

2010-12-29, 01:52	#209
roger Oct 2006 2²×5×13 Posts	Alright, time for some updating after Christmas holidays: 10100<n<11000 k<2.5e7 done. 10100<n<11000 2.5e7<k<5e7 97.8% done [16 hours remaining] 10100<n<11000 5e7<k<1e8 72.6& done. [211 hours remaining] total found: 754. 10000<n<48500 k<1e6 done. total found: 186. 11000<n<48500 1e6<k<5e6 5% done (at n=12890) total found: 141. 123456<n<123476 k<1e7 85% done (74% of n=123474 done) Nothing major to report unfortunately, just a status update. Last fiddled with by roger on 2010-12-29 at 01:53

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