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Old 2007-10-15, 20:37   #1
Jean Penné
 
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Default Even k's and the Sierpinski conjecture

Hi All,
I think I have now a rather complete sight on the problem :

- First, as I posted on the Programming thread, I searched for all even k's, less than 78557, and giving no prime up to n = 4096.

There are 89 such k's.

-secondly, I eliminated all k's that are multiples of the odd values archived by
Ballinger and Keller for the Sierpinski problem ; there are 39 such even k's to drop.

Then 50 k's are remaining

-The multiples of the k values tested and/or eliminated by the Seventeen or Bust project must also be eliminated, there are 17 such values(this is only a coincidence!).

Then 33 even k's are remaining.

-Using the "calc" package, I then eliminated the k's yielding a prime for a n less than 8192. 30 k's are dropped again.

The 3 k's remaining are : 55816, 65536 and 69998

-Using NewPgen and LLR, I found :
55816*2^14536+1 is prime! Time: 358.173 ms.

-I the Chris Caldwell's database, I found :

34999*2^462058+1, 139098 digits, g189, Apr 2001

so, 69998*2^462057+1 is prime!

Only 65536 is remaining, but, as I said, it is special, and I don't recommand to anyone to search for a Fermat prime greater than F4 !

So, no need of a new project on this problem...

But I can say again (if we accept the even k's) :

If F4 is the largest Fermat prime, then, the Sierpinski conjecture is false.

Regards,
Jean
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Old 2007-10-15, 21:48   #2
petrw1
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Quote:
Originally Posted by Jean Penné View Post
Only 65536 is remaining, but, as I said, it is special, and I don't recommand to anyone to search for a Fermat prime greater than F4 !
Pardon me if I misunderstand but is not F4 = 2^(2^4)+1 = 65537 ?
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Old 2007-10-15, 23:36   #3
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Quote:
Originally Posted by Jean Penné View Post
Hi All,
I think I have now a rather complete sight on the problem :

The 3 k's remaining are : 55816, 65536 and 69998

-Using NewPgen and LLR, I found :
55816*2^14536+1 is prime! Time: 358.173 ms.

-I the Chris Caldwell's database, I found :

34999*2^462058+1, 139098 digits, g189, Apr 2001

so, 69998*2^462057+1 is prime!

Only 65536 is remaining, but, as I said, it is special, and I don't recommand to anyone to search for a Fermat prime greater than F4 !

So, no need of a new project on this problem...

But I can say again (if we accept the even k's) :

If F4 is the largest Fermat prime, then, the Sierpinski conjecture is false.

Regards,
Jean
Very nice (and quick!) work. 34999*2^462058+1 was listed on the Ballinger and Keller web-page in the short table beneath the other one, but the other prime 55816*2^14536+1 = 6977*2^14539+1 is not listed in any Sierpinski problem literature because 6977 was eliminated by 6977*2^3+1, but 55816 was still unresolved.

Even if the "odd k Sierpinski conjecture" is eventually settled, the "even k Sierpinski conjecture" may never be!
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Old 2007-10-15, 23:38   #4
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Quote:
Originally Posted by petrw1 View Post
Pardon me if I misunderstand but is not F4 = 2^(2^4)+1 = 65537 ?
Yes it is, Jean's point is that any prime of the form 65536*2^n+1 with n>0 must also be a Fermat prime.
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Old 2007-10-16, 06:21   #5
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Given the success of these 2 problems (riesel and Sierpinski) I am just wondering, if it is possible to have a sierpinski or riesel k such that k*2^n+1 or k*2^n-1 is part of the covering set.

If there is no such k, then every k that produces one prime, will continue to produce more. So if we take a low weight number that produces one small prime, then it will produce more primes eventually.

Any thoughts?
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Old 2007-10-16, 06:26   #6
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Quote:
Originally Posted by petrw1 View Post
Pardon me if I misunderstand but is not F4 = 2^(2^4)+1 = 65537 ?
The Sierpinski and Riesel conjectures require that the n value be a positive integer. Zero is neither positive nor negative.(I could be mistaken about that last sentence. Either way, n is required to be 1 or greater as part of the conjecture)
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Old 2007-10-16, 08:28   #7
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If it isn't too much to ask, can someone prepare an explicit list of all even k's from 2 thru 78556, along with the n value that eliminates it? This would guard against any potential miscalculations.
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Old 2007-10-16, 10:34   #8
Jean Penné
 
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Quote:
Originally Posted by axn1 View Post
If it isn't too much to ask, can someone prepare an explicit list of all even k's from 2 thru 78556, along with the n value that eliminates it? This would guard against any potential miscalculations.
Yes, I will try to do that as soon as possible.
Jean
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Old 2007-10-16, 12:37   #9
Jean Penné
 
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Default Fermat primes and even Sierpinski number

This subject becomes really exciting :

Now, we know that 65536 is the only remaining even Sierpinski candidate able to refute the conjecture, and that :

65536 is a Sierpinski number <---> F4 = 65537 is the largest Fermat prime.

It might be demonstrated if we find a covering set for this even k...

But, also, it is well known that Fermat numbers are pairwise relatively prime, so,

If it exists, the covering set MUST BE INFINITE !!

Interesting, no?

Jean
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Old 2007-10-18, 05:43   #10
Jean Penné
 
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Default The point on the even k's Sierpinski problem

Now, I can make the point on this problem :

1) Eliminating the k's giving a prime for n < 4096 = 2^12

I used a function I wrote for the "calc" package, to compute the "frequencies", as defined by Ray Ballinger and Wilfrid Keller :

; sfreq(0,2,78557,1,2,1)
7205
; sfreq(1,2,78557,1,2,1)
10166
; sfreq(2,2,78557,1,2,1)
9703
; sfreq(3,2,78557,1,2,1)
6204
; sfreq(4,2,78557,1,2,1)
3052
; sfreq(5,2,78557,1,2,1)
1437
; sfreq(6,2,78557,1,2,1)
629
; sfreq(7,2,78557,1,2,1)
351
; sfreq(8,2,78557,1,2,1)
227
; sfreq(9,2,78557,1,2,1)
122
; sfreq(10,2,78557,1,2,1)
55
; sfreq(11,2,78557,1,2,1)
38

78556/2 - 7205 - 10166 - 9703 - 6204 - 3052 - 1437 - 629 - 351 - 227 - 122 - 55 - 38 = 89

89 k's are remaining, whose Keller exponent is at least 2^12

All the (k, n) pairs found by the function are registered in 12 files I will make available on my personal site as soon as possible.

2) 40 k's are eliminated using the archive by Ballinger and Keller :

5794 9714 f13 (2897 9715 R. Baillie, G. Cormack and H. C. Williams, 1981)
6122 33287 f14 (3061 33288 D. A. Buell and J. Young, May 1988)
10594 50010 f15 (5297 50011 J. Young, August 1997)
11588 9713 f13 (2897 9715 R. Baillie, G. Cormack and H. C. Williams, 1981)
11794 22618 f14 (5897 22619 W. Keller, D. A. Buell and J. Young, 1988)
12244 33286 f14 (3061 33288 D. A. Buell and J. Young, May 1988)
14026 126112 f16 (7013 126113 J. Young, August 1997)
15302 25367 f14 (7651 25368 D. A. Buell and J. Young, May 1988)
16846 55156 f15 (8423 55157 D. A. Buell and J. Young, May 1988)
21188 50009 f15 (5297 50011 J. Young, August 1997)
23176 9712 f13 (2897 9715 R. Baillie, G. Cormack and H. C. Williams, 1981)
23588 22617 f14 (5897 22619 W. Keller, D. A. Buell and J. Young, 1988)
24488 33285 f14 (3061 33288 D. A. Buell and J. Young, May 1988)
27574 53134 f15 (13787 53135 J. Young, August 1997)
28052 126111 f16 (7013 126113 J. Young, August 1997)
28054 40638 f15 (14027 40639 D. A. Buell and J. Young, May 1988)
30604 25366 f14 (7651 25368 D. A. Buell and J. Young, May 1988)
33634 42154 f15 (16817 42155 D. A. Buell and J. Young, May 1988)
33692 55155 f15 (8423 55157 D. A. Buell and J. Young, May 1988)
36214 21278 f14 (18107 21279 W. Keller, D. A. Buell and J. Young, 1988)
41702 10671 f13 (20851 10672 W. Keller, D. A. Buell and J. Young, 1988)
42376 50008 f15 (5297 50011 J. Young, August 1997)
46352 9711 f13 (2897 9715 R. Baillie, G. Cormack and H. C. Williams, 1981)
47176 22616 f14 (5897 22619 W. Keller, D. A. Buell and J. Young, 1988)
48976 33284 f14 (3061 33288 D. A. Buell and J. Young, May 1988)
51638 111841 f16 (25819 111842 J. Young, August 1997)
55148 53133 f15 (13787 53135 J. Young, August 1997)
55846 158624 f17 (27923 158625 J. Young, August 1997)
56104 126110 f16 (7013 126113 J. Young, August 1997)
56108 40637 f15 (14027 40639 D. A. Buell and J. Young, May 1988)
61208 25365 f14 (7651 25368 D. A. Buell and J. Young, May 1988)
64322 43795 f15 (32161 43796 D. A. Buell and J. Young, May 1988)
67268 42153 f15 (16817 42155 D. A. Buell and J. Young, May 1988)
67384 55154 f15 (8423 55157 D. A. Buell and J. Young, May 1988)
69422 10463 f13 (34711 10464 W. Keller, D. A. Buell and J. Young, 1988)
69998 462057 f18 (34999 462058 Lew Baxter, 11 Apr 2001)
72428 21277 f14 (18107 21279 W. Keller, D. A. Buell and J. Young, 1988)
73966 38572 f15 (36983 38573 D. A. Buell and J. Young, May 1988)
75122 16603 f14 (37561 16604 W. Keller, D. A. Buell and J. Young, 1988)
78158 26505 f14 (39079 25506 W. Keller, 1988)

49 k's are remaining...

3) 17 k's can be now disregarded thanks to the tremendous work done by the Seveteen or Bust project :

9694 3321062 f21 (4847 3321063 Hassler, SB9, 15 October 2005)
10718 5054501 f22 (5359 5054502 Sundquist, SB6, 6 December 2003)
19388 3321061 f21 (4847 3321063 Hassler, SB9, 15 October 2005)
20446 >10133175 >f22 (10223 >10133176 n upper bound, the 17 October 2007)
21436 5054500 f22 (5359 5054502 Sundquist, SB6, 6 December 2003)
38498 13018585 f23 (19249 13018586 Agafonov, SB10, 5 May 2007)
38776 3321060 f21 (4847 3321063 Hassler, SB9, 15 October 2005)
40892 >10133174 >f22 (10223 >10133176 n upper bound, the 17 October 2007)
42362 >11543658 >f22 (21181 >11543659 n upper bound, the 17 October 2007)
42872 5054499 f22 (5359 5054502 Sundquist, SB6, 6 December 2003)
45398 >12768308 >f22 (22699 >12768309 n upper bound, the 17 October 2007)
49474 >12381605 >f22 (24737 >12381606 n upper bound, the 17 October 2007)
55306 9167432 f23 (27653 9167433 Gordon, SB8, 8 June 2005)
56866 7830456 f22 (28433 7830457 Team Prime Rib, SB7, 30 December 2004)
67322 >11047414 >f22 (33661 >11047415 n upper bound, the 17 October 2007)
76996 13018584 f23 (19249 13018586 Agafonov, SB10, 5 May 2007)
77552 3321059 f21 (4847 3321063 Hassler, SB9, 15 October 2005)

still 32 k's remaining...

4) Eliminating 30 new k's while searching for n up to 8192 using "calc" :

; firstp(766,4094,8192,1)
766*2^6392+1 is a Keller prime! --> (766 6392), (1532 6391), (3064 6390), (6128 6389),
(12256 6388), (24512 6387), (49024 6386)
; firstp(12638,4094,8192,1)
12638*2^4605+1 is a Keller prime! --> (12638 4605), (25276 4604), (50552 4603)
; firstp(14986,4094,8192,1)
14986*2^5248+1 is a Keller prime! --> (14986 5248), (29972 5247), (59944 5246)
; firstp(15914,4094,8192,1)
15914*2^5063+1 is a Keller prime! --> (15914 5063), (31828 5062), (63656 5061)
; firstp(17086,4094,8192,1)
17086*2^5792+1 is a Keller prime! --> (17086 5792), (34172 5791), (68344 5790)
; firstp(32296,4094,8192,1)
32296*2^7560+1 is a Keller prime! --> (32296 7560), (64592 7559)
; firstp(36406,4094,8192,1)
36406*2^6140+1 is a Keller prime! --> (36406 6140), (72812 6139)
; firstp(42334,4094,8192,1)
42334*2^6094+1 is a Keller prime! --> (42334 6094)
; firstp(43606,4094,8192,1)
43606*2^8000+1 is a Keller prime! --> (43606 8000)
; firstp(47558,4094,8192,1)
47558*2^5233+1 is a Keller prime! --> (47558 5233)
; firstp(50678,4094,8192,1)
50678*2^4437+1 is a Keller prime! --> (50678 4437)
; firstp(51722,4094,8192,1)
51722*2^4847+1 is a Keller prime! --> (51722 4847)
; firstp(55816,4094,8192,1)
no 55816*2^n+1 prime for n <= 8192
; firstp(64786,4094,8192,1)
64786*2^4364+1 is a Keller prime! --> (64786 4364)
; firstp(69998,4094,8192,1)
no 69998*2^n+1 prime for n <= 8192
; firstp(73652,4094,8192,1)
73652*2^7455+1 is a Keller prime! --> (73652 7455)
;
30 even k values are eliminated here ; all of them belong to f12.

Only 55816 and 65536 are now remaining

-Using NewPgen and LLR, I found :
55816*2^14536+1 is prime! Time: 358.173 ms. 15 October 2007

As pointed by Philmoore, 55816*2^14536+1 = 6977*2^14539+1 is not listed in any Sierpinski problem literature because 6977 was eliminated by 6977*2^3+1, but 55816 was still unresolved.

So, we may conclude :

-The even k's Sierpinski conjecture is likely to be false...
This would be demonstrated if somebody proves that F4 = 65537 is the largest Fermat prime.

- On the contrary, if it was true...
To prove that would require :

The Seveteen or bust project succeeds to eliminate 5 the odd k's :

10223, 21181, 22699, 24737, 33661

AND Somebody finds a Fermat prime larger than F4 = 65537, thus eliminating k = 65536.

I am thinking that only the first eventuality can be possible...

Regards,
Jean
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Old 2007-10-24, 18:53   #11
Jean Penné
 
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Default Data relevant to the problem

All the (k n) pairs of the Keller primes relevant to this problem are contained in 22 files that can be dowloaded from the URL :

http://jpenne.free.fr/Sierpeven/

Finally only 7 even k's remain unresolved :

20446, 40892, 42362, 45398, 49474, 67322, all bound to the SoB project...

And 65536 = F4 - 1...

Regards,
Jean
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