Bases 2 & 4 reservations/statuses/primes - Page 2

Jean Penné · 2008-01-11, 07:38

Quote:

Originally Posted by gd_barnes

OK, I will make note of the test limits and unreserve k=13854, 16734, and 19464. Note that the last one is currently only on the base 16 page because it's a MOB base 4.

Here is what I show for the test limits for the k's that you're unreserving:

k=13854 and 16734; I show no work so they are still at n=100K base 4.
k=19464 per your status report of 1/7/08 is at n=137112 base 2.

Is that correct?

I'll coordinate getting the 3 unreserved k's tested. If they aren't reserved in the new few days, I'll reserve and test them.

Since you did some testing on k=19464, do you have a sieve file for that one or for any of the k's? If so, I will post them on the website.

Gary

OK for unreserving k=13854 and 16734 ; I am presently sieving for k = 19464, and I wish still test it at least up to 524288 base 2 when sieved enough...

Regards,
Jean

gd_barnes · 2008-01-11, 07:43

Quote:

Originally Posted by Jean Penné

OK for unreserving k=13854 and 16734 ; I am presently sieving for k = 19464, and I wish still test it at least up to 524288 base 2 when sieved enough...

Regards,
Jean

OK, are k=13854 and 16734 still at n=100K base 4?

Gary

Jean Penné · 2008-01-11, 08:49

Quote:

Originally Posted by gd_barnes

OK, are k=13854 and 16734 still at n=100K base 4?

Gary

k = 13854 is at n = 261944 base 2 (so, 128K base 4).

I did'nt worked yet on k = 16734.

Jean

Citrix · 2008-01-13, 04:21

Jean,

I looked at the parity thing you mentioned. It sounds interesting, perhaps we can go to higher bases once we are done with 256.

I also looked at all k's 1-32, that are supper fast for all possible parities up to 2^100. There were some really low weight sequences and some high weight sequences. Do you think we might have similar luck like RPS or 15K with some of these high weight or low weight sequences and possibly find a 10M prime?

What do you think? If you or any one else is interested please let me know.

Thanks

Citrix · 2008-01-13, 05:08

Low weight Reisel
K=29 not included as it is already low weight.

Code:

k*2^m*(2^i)^n-1 // last column is the candidates left, approx estimate of weight
19 9 17 163
19 9 34 332
15 8 35 244
15 28 35 243
19 9 51 496
21 15 55 480
19 2 63 549
21 45 65 416
23 39 65 554
19 43 68 664
21 47 69 613
27 15 69 623
15 1 70 456
15 7 70 384
15 15 70 481
15 21 70 453
15 29 70 483
15 35 70 483
15 43 70 482
15 49 70 483
15 57 70 388
15 63 70 485
15 31 74 457

Jean Penné · 2008-01-13, 07:26

Quote:

Originally Posted by Citrix

Jean,

I looked at the parity thing you mentioned. It sounds interesting, perhaps we can go to higher bases once we are done with 256.

I also looked at all k's 1-32, that are supper fast for all possible parities up to 2^100. There were some really low weight sequences and some high weight sequences. Do you think we might have similar luck like RPS or 15K with some of these high weight or low weight sequences and possibly find a 10M prime?

What do you think? If you or any one else is interested please let me know.

Thanks

Harsh,

Presently, I am more interested in trying to prove one or more of these four mathematical conjectures, than to find large primes (although it may be a subproduct).
Moreover, I hope we will not need to find a 10M prime before proving at least one of them!
So, perhaps it would be better to restrict this project to base 4 for now, and not to dissipate our efforts too much...
But, indeed there is place for other similar projects!

Regards,
Jean

gd_barnes · 2008-01-14, 19:19

Quote:

Originally Posted by Jean Penné

Hi,

On the 23 May 2006, Citrix warned us, in the Sierpinski base 4 thread, about this problem :

http://www.primepuzzles.net/problems/prob_036.htm

To be short, the Liskovets assertion is :

There are some k values such that k*2^n+1 is composite for all n values of certain fixed parity, and some k values such that k*2^n-1 is composite for all n values of certain fixed parity.

It is almost evident that these k values must be searched only amongst the multiples of 3 (the assertion is trivial if 3 does not divide k) :

If k == 1 mod 3, then 3 | k*2^n-1 if n is even, and 3 | k*2^n+1 if n is odd.
If k == 2 mod 3, then 3 | k*2^n+1 if n is even, and 3 | k*2^n-1 if n is odd.

Almost immediately after, Yves Gallot discovered the firt four Liskovets-Gallot numbers ever produced :

k*2^n+1=composite for all n=even: k=66741
k*2^n+1=composite for all n=odd: k=95283
k*2^n-1=composite for all n=even: k=39939
k*2^n-1=composite for all n=odd: k=172677

And Yves said that "I conjecture that 66741, 95283, 39939, ... and 172677 are the smallest solutions for the forms - having no algebraic factorization (such as 4*2^n-1 or 9*2^-1) - but I can't prove it."

For several reasons, I think it would be interesting for us to coordinate the search in order to prove these four conjectures :

1) They involve only k values that are multiples of 3, so the success will no more be depending of the SoB or Rieselsieve one.

2) For the n even Sierpinski case, only k = 23451 and k = 60849 are remaining, with n up to more than 1,900,000 that is to say there are only two big primes to found, then the conjecture is proven!

3) For the n even Riesel (third line above) there are only four k values remaining : 9519, 14361, 19401 and 20049, although the search is only at the beginning!

4) For the two remaining n odd Sierpinski / Riesel (which can be tranlated as
base 4, k even, and doubling the Gallot values : 190566 for k*4^n+1, 345354 for k*4^n-1) I began to explore the problem, by eliminating all k's yielding a prime for n < 4096, eliminating the perfect square k values for Riesel, eliminating the MOB that are redondant, etc...

Finally, there were 42 k values remaining for +1, 114 for -1, and after sieving rapidly with NewPGen, and LLRing up to ~32K, I have now 21 values remaining for +1 and 37 values for -1.

I would be happy to know your opinion about all that...
Regards,
Jean

Jean,

Is there any reason that we are not testing even k's (i.e. multiples of the base) with these conjectures? If a k is even but is not divisible by 4, it yields a different set of factors and prime than any other odd k.

I am testing the Sierp odd-n conjecture of k=95283. Can you tell me how you arrived at 21 k-values remaining at n=32K? I have now tested up to n=56K. I just now finished sieving up to n=200K and am starting LLRing now.

At n=56K, I show 22 odd k's and 8 even k's remaining that are not redundant with other k's remaining; for a total of 30 k's.

At n=32K, I showed 26 odd k's and 9 even k's remaining; for a total of 35 k's.

I checked the top-5000 site for previous smaller primes and there were none for these k's so I wonder why you have less k's remaining than me.

Here are the k's that I show remaining at n=32K, both odd and even, and primes that I found for n=32K-56K for the Sierp odd-n conjecture:

Code:

  k     comments/prime
2943
9267
17937   prime n=53927
24693
26613
29322   even
32247
35787   prime n=36639
37953
38463
39297
43398   even
46623
46902   even
47598   even
50433
53133
60357
60963
61137
61158   even; prime n=48593
62307   prime n=44559
67542   even
67758   even
70467
75183   prime n=35481
78753
80463
83418   even
84363
85287
85434   even
91437
93477
93663

Thanks,
Gary

Jean Penné · 2008-01-14, 21:17

Quote:

Originally Posted by gd_barnes

Jean,

Is there any reason that we are not testing even k's (i.e. multiples of the base) with these conjectures? If a k is even but is not divisible by 4, it yields a different set of factors and prime than any other odd k.

I am testing the Sierp odd-n conjecture of k=95283. Can you tell me how you arrived at 21 k-values remaining at n=32K? I have now tested up to n=56K. I just now finished sieving up to n=200K and am starting LLRing now.

At n=56K, I show 22 odd k's and 8 even k's remaining that are not redundant with other k's remaining; for a total of 30 k's.

At n=32K, I showed 26 odd k's and 9 even k's remaining; for a total of 35 k's.

I checked the top-5000 site for previous smaller primes and there were none for these k's so I wonder why you have less k's remaining than me.

Here are the k's that I show remaining at n=32K, both odd and even, and primes that I found for n=32K-56K for the Sierp odd-n conjecture:

Code:

  k     comments/prime
2943
9267
17937   prime n=53927
24693
26613
29322   even
32247
35787   prime n=36639
37953
38463
39297
43398   even
46623
46902   even
47598   even
50433
53133
60357
60963
61137
61158   even; prime n=48593
62307   prime n=44559
67542   even
67758   even
70467
75183   prime n=35481
78753
80463
83418   even
84363
85287
85434   even
91437
93477
93663

Thanks,
Gary

In the definitions of these four conjectures, the k multipliers must be odd!
For example : 46902*2^n+1 is the same as 23451*2^(n+1)=+1 and if n is odd, n+1 is even, so, you are testing an even exponents candidate!
So, the 8 even k's remaining are relevant to the even n conjecture, and not to the odd n one...

Also, I tested k = 46623 up to n = 79553 and found a prime, so me are almost matching now... I am terminating to gather my results, and will send them to this thread as soon as possible.
Regards,
Jean

gd_barnes · 2008-01-14, 22:42

Quote:

Originally Posted by Jean Penné

In the definitions of these four conjectures, the k multipliers must be odd!
For example : 46902*2^n+1 is the same as 23451*2^(n+1)=+1 and if n is odd, n+1 is even, so, you are testing an even exponents candidate!
So, the 8 even k's remaining are relevant to the even n conjecture, and not to the odd n one...

Also, I tested k = 46623 up to n = 79553 and found a prime, so me are almost matching now... I am terminating to gather my results, and will send them to this thread as soon as possible.
Regards,
Jean

Well, DUH!!

I will remove the even k's that obviously go with the even-n conjecture from my LLRing.

Gary

gd_barnes · 2008-01-15, 18:37

I have now run the Sierp odd-n conjectures up to n=115K and will be continuing on to n=200K sometime next week after completing some sieving for conjectures team drive #1 and a couple of other things. Below are the k's left at n=32K with primes found for n=32K-115K.

I decided to leave the even k's in because in effect it is testing the even conjecture for all k < 95282/2=47641 and I had already sieved them. That should save a lot of effort on that side.

Code:

  k     comments/prime
2943    prime n=108041
9267
17937   prime n=53927
24693
26613   prime n=89749
29322   even; prime n=91367
32247
35787   prime n=36639
37953
38463   prime n=58753
39297
43398   even; prime n=72873
46623   prime n=79553
46902   even
47598   even; prime n=105899
50433
53133
60357
60963   prime n=73409
61137
61158   even; prime n=48593
62307   prime n=44559
67542   even
67758   even
70467
75183   prime n=35481
78753   prime n=63761
80463
83418   even; prime n=80593
84363
85287
85434   even
91437
93477   prime n=63251
93663   prime n=82317

Total of 14 odd k's and 4 even k's remaining.

So...based on this effort by itself, here are the statuses of the base 2 Sierp odd-n and even-n cojectures:

Odd-n:
14 k's remaining at n=115K from odd k's above. k's remaining:
9267
24693
32247
37953
39297
50433
53133
60357
61137
70467
80463
84363
85287
91437

Even-n:
47641<k<66741: still needs to be tested.
k<=47641: 4 k's remaining at n=115K from even k's above. k's remaining converted to odd-k:
23451
33771
33879
42717

Edit: I just now realized that it was already stated that only k=23451 and 60849 are remaining on the even-n side as a result of the Sierp base 4 project. OK, NEXT time I'll remove the even k's from my testing. Ergh!

Gary

Jean Penné · 2008-01-15, 20:28

Gary,

I am gathering my results about the four conjectures, which requires a lot of work...
So, we will be able to compare with your results!

Here for +1 and n odd : There are 23 remaining candidates, and 42 primes found.

1) Remaining :

Code:

  k       2k       tested up to (n-1 base 2)

9267    18534         1967862
32247   64494         1770506
37953   75906           33448
38463   76926           34320
39297   78594           35166
50433   100866          46076
53133   106266          87428
56643   113286          33348
60357   120714          46166
60963   121926          32912
61137   122274          33150
62307   124614          35342
70467   140934          46358
75183   150366          32840
78153   156306          32976
78483   156966          33096
78753   157506          55640
80463   160926          35660
84363   168726          35008
85287   170574          33106
91437   182874          33034
93477   186954          34846
93663   187326          67844

2) Primes :

Code:

Normalized	As discovered
 k         n
93       20917	186*2^20916+1 is prime!  Time: 786.013 ms.
2943    108041	5886*2^108040+1 is prime! by Jean Penné 06/02/05, 09:28AM
5193      4277	10386*2^4276+1 is prime!  Time: 59.124 ms.
5703      5149	11406*2^5148+1 is prime!  Time: 99.286 ms.
5823      8105	11646*2^8104+1 is prime!  Time: 384.321 ms.
6807      4415	13614*2^4414+1 is prime!  Time: 64.834 ms.
6843     14753	13686*2^14752+1 is prime!  Time: 935.778 ms.
7233      4277	14466*2^4276+1 is prime!  Time: 58.406 ms.
9777     18975	19554*2^18974+1 is prime!  Time: 791.931 ms.
10923     6801	21846*2^6800+1 is prime!  Time: 177.623 ms.
14397     4347	28794*2^4346+1 is prime!  Time: 59.056 ms.
16917    12799	33834*2^12798+1 is prime!  Time: 829.050 ms.
17457    29563	34914*2^29562+1 is prime! by tcadigan 28/01/05, 07:13AM
17937    53927	35874*2^53926+1 is prime! by tcadigan 28/01/05, 06:41AM
20997     8191	41994*2^8190+1 is prime!  Time: 384.323 ms.
22653    28969	45306*2^28968+1 is prime! by Mark 27/01/05, 11:29PM
24693   357417	49386*2^357416+1 is Prime! by Footmaster 25/05/05, 08:44AM
25083    24981	50166*2^24980+1 is prime! by Mark 27/01/05, 11:29PM
25917     9671	51834*2^9670+1 is prime!  Time: 447.572 ms.
26613    89749	53226*2^89748+1 is prime!  Time: 25.000 sec.
30933     4433	61866*2^4432+1 is prime!  Time: 59.831 ms.
35787    36639	71574*2^36638+1 is prime!  Time: 6.016 sec.
40857     5383	81714*2^5382+1 is prime!  Time: 154.925 ms.
42993    16165	85986*2^16164+1 is prime!  Time: 1.223 sec.
43167     9795	86334*2^9794+1 is prime!  Time: 403.492 ms.
46623    79553	93246*2^79552+1 is prime!  Time: 28.184 sec.
49563     5813	99126*2^5812+1 is prime!  Time: 186.733 ms.
60273     7421	120546*2^7420+1 is prime!  Time: 218.279 ms.
63357     4211	126714*2^4210+1 is prime!  Time: 136.667 ms.
65223     4189	130446*2^4188+1 is prime!  Time: 136.782 ms.
65253    10301	130506*2^10300+1 is prime!  Time: 427.195 ms.
67917    13079	135834*2^13078+1 is prime!  Time: 644.241 ms.
69963     5205	139926*2^5204+1 is prime!  Time: 152.156 ms.
72537    15771	145074*2^15770+1 is prime!  Time: 1.199 sec.
73023    17965	146046*2^17964+1 is prime!  Time: 1.349 sec.
78543    10089	157086*2^10088+1 is prime!  Time: 419.114 ms.
80517     5423	161034*2^5422+1 is prime!  Time: 154.232 ms.
81147    17615	162294*2^17614+1 is prime!  Time: 1.331 sec.
82197     5079	164394*2^5078+1 is prime!  Time: 149.421 ms.
88863     9825	177726*2^9824+1 is prime!  Time: 405.011 ms.
91383    15333	182766*2^15332+1 is prime!  Time: 1.173 sec.
93033    30473	186066*2^30472+1 is prime!  Time: 5.037 sec.

Note : The name of the discoverer is shown only for primes found by the Sierpinski base 4 project.

Regards,
Jean

2008-01-13, 04:21	#15
Citrix Jun 2003 3×23² Posts	Jean, I looked at the parity thing you mentioned. It sounds interesting, perhaps we can go to higher bases once we are done with 256. I also looked at all k's 1-32, that are supper fast for all possible parities up to 2^100. There were some really low weight sequences and some high weight sequences. Do you think we might have similar luck like RPS or 15K with some of these high weight or low weight sequences and possibly find a 10M prime? What do you think? If you or any one else is interested please let me know. Thanks Last fiddled with by Citrix on 2008-01-13 at 04:22

2008-01-15, 18:37	#21
gd_barnes May 2007 Kansas; USA 10100111001100₂ Posts	I have now run the Sierp odd-n conjectures up to n=115K and will be continuing on to n=200K sometime next week after completing some sieving for conjectures team drive #1 and a couple of other things. Below are the k's left at n=32K with primes found for n=32K-115K. I decided to leave the even k's in because in effect it is testing the even conjecture for all k < 95282/2=47641 and I had already sieved them. That should save a lot of effort on that side. Code: k comments/prime 2943 prime n=108041 9267 17937 prime n=53927 24693 26613 prime n=89749 29322 even; prime n=91367 32247 35787 prime n=36639 37953 38463 prime n=58753 39297 43398 even; prime n=72873 46623 prime n=79553 46902 even 47598 even; prime n=105899 50433 53133 60357 60963 prime n=73409 61137 61158 even; prime n=48593 62307 prime n=44559 67542 even 67758 even 70467 75183 prime n=35481 78753 prime n=63761 80463 83418 even; prime n=80593 84363 85287 85434 even 91437 93477 prime n=63251 93663 prime n=82317 Total of 14 odd k's and 4 even k's remaining. So...based on this effort by itself, here are the statuses of the base 2 Sierp odd-n and even-n cojectures: Odd-n: 14 k's remaining at n=115K from odd k's above. k's remaining: 9267 24693 32247 37953 39297 50433 53133 60357 61137 70467 80463 84363 85287 91437 Even-n: 47641<k<66741: still needs to be tested. k<=47641: 4 k's remaining at n=115K from even k's above. k's remaining converted to odd-k: 23451 33771 33879 42717 Edit: I just now realized that it was already stated that only k=23451 and 60849 are remaining on the even-n side as a result of the Sierp base 4 project. OK, NEXT time I'll remove the even k's from my testing. Ergh! Gary Last fiddled with by gd_barnes on 2008-01-15 at 18:47

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