Very Prime Riesel and Sierpinski k - Page 50

Thomas11 · 2013-07-29, 14:16

Quote:

Originally Posted by robert44444uk

I thought I would plot x=nth prime versus y=ln2(ln2(k*2^n-1)) and see what I got. As expected a very linear relationship and what's more this suggests the 200th prime at about n=475,000 unless I did the maths wrong. Actually, I think it will be greater than this value.

Seems that your estimate was right on the spot!
At least until you will not discover another prime below n=250k.

robert44444uk · 2013-07-30, 08:28

Quote:

Originally Posted by robert44444uk

I thought I would plot x=nth prime versus y=ln2(ln2(k*2^n-1)) and see what I got. As expected a very linear relationship and what's more this suggests the 200th prime at about n=475,000 unless I did the maths wrong.

Gosh, what was I thinking when I typed that formula! I meant:

x=nth prime versus y=log2(log2(k*2^n-1)) where log2 is log base 2.

Starting this morning at n=227953. I should be able to speed up when I get to 230000.

robert44444uk · 2013-07-30, 13:01

I did some factoring to try to understand why the candidate is so prime. The results are interesting:

odd primes <10000 total 1,228
Primes <10000 with modulo2 <1001 total 433

Looking at the 433 primes:

A Payam E(130) to which this candidate belongs eliminates 15 primes <10000 where prime p has modulo2 of p-1, and 86 other primes, a total of 101 primes <10000
There are 52 primes >131 and <1000 that are p-1 modulo2 (these cannot be eliminated in a power series base 2)

This leaves 280 primes that are modulo 2 <1000 where a Payam number might be more efficient than another Payam number of the same E(130) class. The 280 is equal to 433-101-52

Of the 280 primes, only 74 of those provide factors in the series, an extremely low number. The candidate eliminates a further 206 primes that are <10000 and are <1000 modulo 2.

These are shown in the list:

Code:

prime    modulo2

271	135
311	155
313	156
383	191
449	224
487	243
577	144
593	148
599	299
607	303
617	154
647	323
691	230
733	244
739	246
751	375
809	404
811	270
823	411
839	419
863	431
919	153
929	464
971	194
983	491
991	495
997	332
1009	504
1021	340
1049	262
1051	350
1063	531
1097	274
1151	575
1153	288
1193	298
1201	300
1217	152
1223	611
1231	615
1249	156
1279	639
1289	161
1327	221
1361	680
1399	233
1433	179
1459	486
1543	771
1553	194
1567	783
1583	791
1597	532
1601	400
1607	803
1609	201
1627	542
1663	831
1697	848
1699	566
1709	244
1721	215
1753	146
1759	879
1811	362
1831	305
1871	935
1889	472
1933	644
1951	975
1993	996
1999	333
2003	286
2017	336
2129	532
2179	726
2203	734
2251	750
2281	190
2287	381
2341	780
2347	782
2383	397
2393	598
2441	305
2473	618
2657	166
2671	445
2689	224
2749	916
2767	461
2791	465
2917	972
2969	371
3049	762
3061	204
3109	444
3121	156
3137	784
3217	804
3221	644
3271	545
3331	222
3343	557
3361	168
3389	484
3449	431
3457	576
3463	577
3529	882
3541	236
3631	605
3739	534
3761	188
3821	764
3823	637
3833	958
3889	648
3943	219
4001	1000
4049	506
4057	169
4129	688
4153	346
4201	525
4211	842
4271	305
4273	534
4297	537
4409	551
4421	340
4423	737
4519	753
4523	266
4567	761
4643	422
4657	388
4663	777
4721	295
4729	788
4733	364
4751	475
4759	793
4861	972
4871	487
4931	986
4993	624
4999	357
5081	635
5167	861
5209	217
5237	748
5279	377
5297	662
5347	198
5441	544
5471	547
5503	917
5531	790
5569	464
5669	436
5689	711
5711	571
5737	239
5821	388
5867	838
5953	992
6043	318
6089	761
6337	288
6353	397
6449	806
6481	810
6563	386
6679	159
6689	836
6857	857
6871	687
6959	497
7001	500
7057	392
7151	325
7351	525
7393	264
7487	197
7489	468
7753	323
7867	874
7993	999
8161	408
8317	308
8353	464
8581	660
8713	363
8761	365
8831	883
8929	496
9041	904
9413	724
9431	943
9511	317
9521	476
9623	283
9721	810
9781	652
9929	292

robert44444uk · 2013-07-30, 14:04

The lack of small factors clearly has large impact on any sieving. The candidate exhibits the following persistence:

Code:

Sieve limit   candidates remaining

10^4           61.02%
10^5           49.29%
10^6           41.09%
10^7           35.09%
6.34*10^12     19.26%

I can't calculate the Nash weight using the software I have.

Thomas11 · 2013-07-30, 15:56

Quote:

Originally Posted by robert44444uk

I can't calculate the Nash weight using the software I have.

Please try the attached program (Windows 32 bit console application).

Usage is a follows:

Code:

Nash_Payam.exe <S/R> <E> <y>

For example:

Code:

Nash_Payam.exe R 130 22544089918041953

R 130 22544089918041953 8818

robert44444uk · 2013-07-30, 16:26

Hah, there is a surprise.

It turns out that the candidate has one of the weakest weights in its class of Riesel E130 Payams that are VPS.

A survey of 189 such candidates has a top weight of 9284, a lowest weight of 8565, and a median weight of 8951 compared to the candidates weight of 8818.

Thank you for lending me a new toy though, Thomas11, I shall play with this for all of the Riesel VPS

As of 17:30 tonight I am up to 228639, and no more prp/primes. I'm still advancing 230-240 and 240-250 on two other consoles. I have checked 199 prp and confirm that they are prime.

robert44444uk · 2013-07-30, 17:16

What lovely toy. I put all of my VPS through it and produced the following:

Code:

	Max	Min	Median
28	7190	6977	6989
36	7776	6878	7538
52	8405	7117	7799
58	8417	7089	7927.5
60	8661	7572	8087
66	8971	7475	8197
82	8870	7906	8419.5
100	9156	8244	8716
106	9196	8418	8817.5
130	9284	8565	8951
138	9344	8705	9068
148	9394	8915	9152
162	9595	9120	9324
172	9497	9357	9418
178	9527	9322	9447
180	9685	9343	9522.5
196	9651	9649	9650

The game does not work for the supergiants - R268 and R292 both show all at 10000

Thomas11 · 2013-07-30, 17:46

Quote:

Originally Posted by robert44444uk

The game does not work for the supergiants - R268 and R292 both show all at 10000

Sorry, it was limited to E<268.

Please try the attached version. It should work up to E=466 (if ever needed).

Thomas11 · 2013-07-30, 20:04

Here is a similar table of the Nash weights for the Sierpinski side:

Code:

 E	count	Min	Max	Average
---------------------------------------
 28	    6	6966	7579	7324.7
 36	    0	 --	 --	 --
 52	 3206	6845	8409	7809.6
 58	 3296	7161	8603	7939.6
 60	 2938	7259	8819	8059.4
 66	 3135	7350	8836	8186.9
 82	 3316	7834	8999	8423.3
100	 1369	8067	9164	8710.7
106	  897	8376	9224	8817.9
130	  856	8562	9312	8949.1
138	  646	8680	9404	9067.5
148	  246	8827	9536	9157.3
162	   42	9108	9525	9315.4
172	   47	9108	9574	9389.6
178	   36	9318	9663	9457.9
180	   42	9285	9734	9520.9
196	    5	9608	9869	9748.2
210	    4	9613	9788	9703
226	    1	9794	9794	9794
---------------------------------------
Total:	20088

Note the similarities of the average weights for Riesel and Sierpinski side!

Thomas11 · 2013-07-31, 07:51

Let's start the day with another prime:

202? 496187

robert44444uk · 2013-07-31, 09:11

Quote:

Originally Posted by Thomas11

Let's start the day with another prime:

202? 496187

The way one does! Sigh. Might get to 230K today.

Robert

2013-07-30, 14:04	#543
robert44444uk Jun 2003 Suva, Fiji 2³×3×5×17 Posts	The lack of small factors clearly has large impact on any sieving. The candidate exhibits the following persistence: Code: Sieve limit candidates remaining 10^4 61.02% 10^5 49.29% 10^6 41.09% 10^7 35.09% 6.34*10^12 19.26% I can't calculate the Nash weight using the software I have.

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2013-07-30, 16:26	#545
robert44444uk Jun 2003 Suva, Fiji 2³·3·5·17 Posts	Hah, there is a surprise. It turns out that the candidate has one of the weakest weights in its class of Riesel E130 Payams that are VPS. A survey of 189 such candidates has a top weight of 9284, a lowest weight of 8565, and a median weight of 8951 compared to the candidates weight of 8818. Thank you for lending me a new toy though, Thomas11, I shall play with this for all of the Riesel VPS As of 17:30 tonight I am up to 228639, and no more prp/primes. I'm still advancing 230-240 and 240-250 on two other consoles. I have checked 199 prp and confirm that they are prime.

2013-07-31, 07:51	#549
Thomas11 Feb 2003 2⁷·3·5 Posts	Let's start the day with another prime: 202? 496187