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-   -   Bases 2 & 4 reservations/statuses/primes (https://www.mersenneforum.org/showthread.php?t=9830)

 Jean Penné 2008-01-11 07:38

[QUOTE=gd_barnes;122588]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[/QUOTE]

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=Jean Penné;122607]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[/quote]

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

Gary

 Jean Penné 2008-01-11 08:49

[QUOTE=gd_barnes;122608]OK, are k=13854 and 16734 still at n=100K base 4?

Gary[/QUOTE]

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:smile:

 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
[/code]

 Jean Penné 2008-01-13 07:26

[QUOTE=Citrix;122746]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:smile:[/QUOTE]

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=Jean Penné;122480]Hi,

[URL]http://www.primepuzzles.net/problems/prob_036.htm[/URL]

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.

Regards,
Jean[/quote]

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]
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
[/code]

Thanks,
Gary

 Jean Penné 2008-01-14 21:17

[QUOTE=gd_barnes;122833]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]
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
[/code]

Thanks,
Gary[/QUOTE]

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=Jean Penné;122841]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[/quote]

Well, DUH!! :blush: I will remove the even k's that obviously go with the even-n conjecture from my LLRing. :rolleyes:

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]
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
[/code]

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

Gathered results for k*2^n+1, n odd

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
[/CODE]

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.
[/CODE]

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

Regards,
Jean

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