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-   -   Sierpinski / Riesel - Base 22 (https://www.mersenneforum.org/showthread.php?t=6916)

michaf 2007-01-09 22:22

Sierpinski / Riesel - Base 22
 
Sierpinski / Riesel - Base 22

Conjectured Sierpinski at 6694 [5,23,97]
Conjectured Riesel at Riesel 4461 [5,23,97]

Pesky 17 k's include (now 13 to go)

Sierpinski:

22 (cedricvonck)
484 (cedricvonck)
1611 (michaf tested upto 12000)
1908 (michaf tested upto 12000)
4233 (michaf tested upto 12000)
5128 (michaf tested upto 12000)
5659 (michaf tested upto 12000)
6462 (michaf tested upto 12000)

Riesel:

1013 (michaf tested upto 12000)
2853 (michaf tested upto 12000)
3104 (michaf tested upto 12000)
3656 (michaf tested upto 12000)
4001 (michaf tested upto 12000)

22 and 484 are special cases; only non-trivials occur with n=2^m
If a prime is found for 22 case, 484 is also eliminated (n is one lower in that case)

(larger) primes found:
4118*22^12347-1 (michaf)
6234*22^16010+1 (michaf)
942*22^18359+1 (michaf)
5061*22^24048+1 (michaf)

22*22^n+1 / 484*22^n+1 status:
[code]
below (512): proven composite with phrot
(512) 22^512+1 has factor 115443366400367617
(1k) 22^1024+1 has factor 2095383775764481
(2k) 22^2048+1 has factor 65465822271579614082713282973697
(4k) 22^4096+1 has factor 40961
(8k) 22^8192+1 has factor 147457
(16k) 22^16384+1 has factor 2342241402881
(32k) 22^32768+1 has factor 65537
(64k) 22^65536+1 has factor 27918337
(128k) 22^131072+1 has factor 786433
(256k) 22^262144+1 has factor 29884417
(512k) 22^524288+1 has factor 93067411457
[B](1M)[/B] no factors upto 1607651162167705601 (also P-1 stage 1 done with B1=100000 and 25 ECM curves, B1=1000, B2=100000)
[B](2M)[/B] no factors upto 285159626880581633
[B](4M)[/B] no factors upto 556968483053633537
[B](8M)[/B] no factors upto 9221584136710389761 (stopping here)
(16M) 22^16777216+1 has factor 189162539974657
(32M) 22^33554432+1 has factor 21096518178045953
[B](64M)[/B] no factor upto 9202942106325221377 (stopping here)
(128M) 22^134217728+1 has factor 91268055041
(256M) 22^268435456+1 has factor 7368180622415626241
[B](512M)[/B] no factor upto 9187751282130026497 (stopping here)
[B](1G)[/B] no factor upto 9159662798383349761 (stopping here)
[/code]

ValerieVonck 2007-01-10 07:24

base 22
 
1 Attachment(s)
n = 22 tested to 144895

fatphil 2007-01-10 16:27

[QUOTE=CedricVonck;95703]n = 22 tested to 144895[/QUOTE]

Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.

Xentar 2007-01-10 18:55

[QUOTE=fatphil;95745]Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.[/QUOTE]

eh.. is it the same with
18 * 18^n+1 ?
and if not, could you please try to explain it to me? (Yea, I'm trying to learn the math behind these projects :wink: )

fatphil 2007-01-10 19:13

[QUOTE=Xentar;95761]eh.. is it the same with
18 * 18^n+1 ?
and if not, could you please try to explain it to me? (Yea, I'm trying to learn the math behind these projects :wink: )[/QUOTE]

They are also just 18^n+1. Again, they can only be prime if n is a power of 2, and would be Generalised Fermat Numbers. Have a look at Yves Gallot's GFN page for more info.

michaf 2007-01-10 22:32

4118*22^12347-1 is prime
6234*22^16010+1 is prime

tests now at 18000

geoff 2007-01-11 05:43

If you want to sieve the GFN's, srsieve should recognise them as such and use a faster method of sieving as long as only the terms of the form n=2^m+1 appear in the input file. If you start srsieve with the -v option it will print a message something like `filtering for primes of the form (2^m)x+1' if it has recognised the sequence as GFN. (If there are non-GFN sequences in the sieve as well then it will still notice the GFN's, but most of the benefit will be lost).

ValerieVonck 2007-01-11 11:18

[QUOTE=fatphil;95745]Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.[/QUOTE]

Apperantely not then... Like I said I used NewPGen to sieve the file:
[code]
k*b^n-1 with k fixed
k = 22
b = 22
n from 2 to 1.000.000
[/code]

Then I fired up LLR.exe (3.7.1)

michaf 2007-01-12 19:23

942*22^18359+1 is prime
5061*22^24048+1 is prime

jasong 2007-01-13 03:05

In terms of 22^(2^m)+1, is it possible to figure the odds that at least one value of m will yield a prime for the equation? Since the numbers get big so quickly, I'm thinking it's possible this may be a Sierpinski number that may never be proven, or at least not within my lifetime.

Can anybody do the math and figure the chance that it would or wouldn't yield prime if every single m value could be figured at once for the above equation?

michaf 2007-01-13 09:32

The chances are quite slim I suspect,
but sieving can be done quite easily.

I have started yesterday a sieve for the first 4 numbers left, and that one did 40Mp/sec (now only 3 left since a factor was found for 512k)

It will reach the limit of srsieve in about 1 day.
If some numbers are left, well... it's getting very hard to test them.

I will start a sieve for the next 4 after the first one finishes; that sieve-speed is significantly slower (About 7Mp/sec), but will finish in reasonable time too.
After that, I think I will let it go.

michaf 2007-01-19 19:06

small succes:

91268055041 | 22^134217728+1

So at least THAT one isn't prime :)

fatphil 2007-01-20 00:43

[QUOTE=michaf;96539]small succes:

91268055041 | 22^134217728+1

So at least THAT one isn't prime :)[/QUOTE]

That's about one millionth of a second's work. Why is it considered a success?

jasong 2007-01-20 03:23

[QUOTE=fatphil;96571]That's about one millionth of a second's work. Why is it considered a success?[/QUOTE]
He's probably thinking of how long it would take to test the number for primality if it hadn't been sieved.

michaf 2007-01-20 08:03

[QUOTE=jasong;96577]He's probably thinking of how long it would take to test the number for primality if it hadn't been sieved.[/QUOTE]

That was indeed what I was thinking :)
_and_ it took some 24 hours to get to that sieving point :>

Citrix 2007-01-20 08:20

[QUOTE=michaf;96590]That was indeed what I was thinking :)
_and_ it took some 24 hours to get to that sieving point :>[/QUOTE]

Should have taken you less than a sec!:smile:

Anyway why are you trying to factorize these numbers
For base 22, numbers that are multiple of 22 do not have to be tested, this eliminates k=22,484 but k=1 is left. But 1*22^1+1 is prime hence 1 is eliminated.

under base=100 only the following bases are left for which k does not produce a prime
38
50
62
68
86
92
98
You can try to find a prime for them

michaf 2007-01-20 18:29

[quote]
Originally Posted by michaf View Post
Without any math skills... so excuse me if I bugger here :>

base 22:

22*22^n + 1
=
22^(n+1) + 1
=
1*22^(n+1) + 1

so, k = 1
and that one is eliminated, therefore is k=22 and 484?[/quote]

[quote]
Unfortunately not. 1*22^1+1=23 prime, but, I think we decided for the Sierpinski base 5 exercise, that we would not use n=0, otherwise k=22 could be eliminated but not 484.[/quote]

(quotes from sierpinski 6-18 thread)
I think this justifies the search

michaf 2007-01-20 23:24

[QUOTE=Citrix;96592]Should have taken you less than a sec!:smile:
[/QUOTE]

Oh, how do you do that? Test for all factors upto that limit in a sec? :)
Or did you mean just testing if that one number divided the huge number?

Citrix 2007-01-21 02:34

[QUOTE=michaf;96643]Oh, how do you do that? Test for all factors upto that limit in a sec? :)
Or did you mean just testing if that one number divided the huge number?[/QUOTE]

Finding the factor should take less than 1 sec. Try to factor 1 number at a time. What numbers are left, I can try to prove them composite them for you.

michaf 2007-01-21 09:58

Hmm... what am I doing wrong then?

srsieve is sieving about 20-30Million p's per second, but not a huge amount more when sieving only 1 n.
Or is srsieve the wrong program here?

Citrix 2007-01-21 11:11

[QUOTE=michaf;96674]Hmm... what am I doing wrong then?

srsieve is sieving about 20-30Million p's per second, but not a huge amount more when sieving only 1 n.
Or is srsieve the wrong program here?[/QUOTE]

Try PFGW.

michaf 2007-01-21 12:54

[QUOTE=Citrix;96679]Try PFGW.[/QUOTE]

I couldn't notice anything quicker about pfgw's factoring routines then there is in srsieve (quite the opposite, actually).

What's pfgw's command to find "91268055041 | 22^134217728+1" quickly?

Citrix 2007-01-21 13:10

[QUOTE=michaf;96683]I couldn't notice anything quicker about pfgw's factoring routines then there is in srsieve (quite the opposite, actually).

What's pfgw's command to find "91268055041 | 22^134217728+1" quickly?[/QUOTE]

Try this
[url]http://www.underbakke.com/AthGFNsv/[/url]

For PFGW see the pfgw documentation.

R. Gerbicz 2007-01-21 13:10

[QUOTE=michaf;96683]I couldn't notice anything quicker about pfgw's factoring routines then there is in srsieve (quite the opposite, actually).

What's pfgw's command to find "91268055041 | 22^134217728+1" quickly?[/QUOTE]

If p is a prime divisor of b^(2^n)+1 then p=2 or p=k*2^(n+1)+1 (where k is an integer).
Using this you can get a much faster sieve.

axn 2007-01-21 13:15

And srsieve (as well as PFGW) has a modular sieve option that will restrict itself to factors of the correct form. [Not sure if sr2sieve has this option implemented]

michaf 2007-01-21 22:26

[QUOTE=Citrix;96685]Try this
[url]http://www.underbakke.com/AthGFNsv/[/url][/quote]
wooot... WOW :>
Much heat has been generated before I was pointed at this program :>

Thanks... this is amazingly quick!

michaf 2007-01-22 19:33

Citrix,

do you happen to know if athGFNSieve can sieve beyond about 9159662798383349761?
Two instances I've had upto that point, and then give up on me. Process is still using CPU power, but no more updates either on screen or in file.

Citrix 2007-01-23 00:47

YOu will have to try prime95 for p-1/ECM after that limit is reached.

michaf 2007-01-23 15:08

Any idea on what good B1, B2 values might be for P1?
I cannot do anything on stage 2 (insufficient memory).

What is usually needed to get say 30 digit factors?

I know how ecm works with that regard, but no idea on what to do with P-1

gd_barnes 2007-11-22 06:41

Eliminate k=22 & 484 and n=0
 
[quote=michaf;96590]That was indeed what I was thinking :)
_and_ it took some 24 hours to get to that sieving point :>[/quote]

We do not include k's that can be reduced when determining the Riesel and Serpinksi conjectures. I am currently compiling a site of all known primes of the form k*10^n-1 at [URL="http://gbarnes017.googlepages.com/primes-kx10n-1.htm"]gbarnes017.googlepages.com/primes-kx10n-1.htm[/URL]. I would never include multiples of 10 for k's because they can always be reduced to (k/10)*10^(n+1)-1. For the same reason, RPS does not include multiples of 2 for k's base 2. If we included multiples of 10 for base 10, we'd still have no primes for k=450 and k=4500 because there is only one prime at n=1 for k=45. (Riesel k base 10 is k=10176. 3 k's still need primes to prove conjecture.)

Also, don't include n=0 when proving these types of conjectures. n=0 is prime for the same k's in all bases and hence is redundant and unnecessary.

As Citrix stated, eliminate k=22 and k=484 from your search because they are multiples of 22. 484*22^n+1 would simply be 22^(n+2)+1. As stated by Phil, k=22 can only be prime if n is a power of 2. The same of course applies to any k that is a power of 22. And any k that is a multiple of 22 can be reduced...i.e. 44*22^n+1 = 2*22^(n+1)+1.

The conjecture is proven if a prime is found for all k's < then the first known Sierpinski # that are both: (1) Not multiples of the base. (2) Do not have the same factor for every occurrence of n. (i.e. every n has a factor of 3 for all k==2 mod 3 for base 22 so clearly you wouldn't count those as a Sierpinski #'s).


Gary

masser 2007-11-22 15:35

Please see

[url]http://www.mersenneforum.org/showthread.php?t=4832[/url]

- the issue of whether n=0 should be counted or not is layed out very well there.

... and don't dictate what others should do.

Siemelink 2007-11-24 19:59

Michaf, are you still working on this numbers? If not, do you mind if I take over?
In the thread there is a post stating that you tested all until n = 18000.
How far did you get?

Willem.

gd_barnes 2007-11-24 23:37

[quote=masser;118979]Please see

[URL]http://www.mersenneforum.org/showthread.php?t=4832[/URL]

- the issue of whether n=0 should be counted or not is layed out very well there.

... and don't dictate what others should do.[/quote]


I'm sorry, Masser, Michaf, and everyone else. Masser was right. I came across as a bit of a 'dictator' there. :sad: (Well, actually a LOT of a dictator.)

Masser, I checked out the thread that you posted here. It is a good one but I guess I am a little bit confused now. In the post, doesn't Geoff state the following?:

[quote]
However, perhaps we need a new definition? We could extend the pattern of definitions above in this way:

2'. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n >= -1.

2''. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n >= -2.

and so on. This leads to the obvious definition:

3. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n.

I think this definition better fits your idea that only candidates that determine unique sequences should be considered.

For the purposes of this project it makes no difference whether we use definition 2 or definition 3.
[/quote]


I can only speculate and maybe you can confirm that we want to use the above base 5 defintion for all bases. Is that your thinking? If so, here is where I'm confused:

Based on this definition, wouldn't we eliminate k=22 from testing for the base-22 Sierpinski conjecture because 22*22^0+1=23, which of course is prime? And wouldn't it also eliminate k=484? Because 484*22^(-1)+1=23, which is prime again?

I'm asking because I saw a suggestion in the 'base 6 to 18' thread that we need one place where all of the information is brought together for the conjectures for all bases. What I would like to do is put together a web page solely dedicated to showing all of the bases (perhaps up to 50 or so), their respective known conjectured Riesel and Sierpinski numbers, and what k's are left to find a prime to prove the conjectures for each base (for the reasonable ones, obv not base 7). If I can understand this k=22 and k=484 issue for base 22, that would be very helpful.


Thank you,
Gary

michaf 2007-11-25 10:01

You can take them,

I cannot get to the computer I tested them on right now, but my guess is that I updated the web-page when I stopped searching.
Good luck!

and @Gary:
apologies accepted, I think the comments were made with the best intentions :)

As for including or not: You might want to just a side-note telling the story, it'll be clear to everyone then.


[QUOTE=Siemelink;119113]Michaf, are you still working on this numbers? If not, do you mind if I take over?
In the thread there is a post stating that you tested all until n = 18000.
How far did you get?

Willem.[/QUOTE]

gd_barnes 2007-11-26 05:21

[quote=michaf;119155]and @Gary:
apologies accepted, I think the comments were made with the best intentions :)

As for including or not: You might want to just a side-note telling the story, it'll be clear to everyone then.[/quote]

OK, thanks for the input, Michaf. In a list of remaining k's to find prime for each base, if there are "exception-situation" k's like this, I will show those with an asterisk by them -or- may not show them but instead put a special note at the bottom of the page about them.

As I get into creating the page, I'll ask people's input about what they think looks the best.


Gary

Siemelink 2007-12-05 20:16

Found some.
 
Aloha everyone.

I have some results:
5659*22^97758+1 is probable prime
6462*22^45507+1 is probable prime
1013*22^26067-1 is probable prime
2853*22^27975-1 is probable prime
4001*22^36614-1 is probable prime

Having fun, Willem.
--
Sierpinski / Riesel - Base 22

Conjectured Sierpinski at 6694 [5,23,97]
Conjectured Riesel at Riesel 4461 [5,23,97]

Sierpinski
22 (cedricvonck)
484 (cedricvonck)
1611 (Willem tested upto 98000)
1908 (Willem tested upto 60000)
4233 (Willem tested upto 30000)
5128 (Willem tested upto 30000)

Riesel
3104 (willem tested upto 60000)
3656 (willem tested upto 60000)

gd_barnes 2007-12-06 03:00

[quote=Siemelink;119977]Aloha everyone.

I have some results:
5659*22^97758+1 is probable prime
6462*22^45507+1 is probable prime
1013*22^26067-1 is probable prime
2853*22^27975-1 is probable prime
4001*22^36614-1 is probable prime

Having fun, Willem.
--
Sierpinski / Riesel - Base 22

Conjectured Sierpinski at 6694 [5,23,97]
Conjectured Riesel at Riesel 4461 [5,23,97]

Sierpinski
22 (cedricvonck)
484 (cedricvonck)
1611 (Willem tested upto 98000)
1908 (Willem tested upto 60000)
4233 (Willem tested upto 30000)
5128 (Willem tested upto 30000)

Riesel
3104 (willem tested upto 60000)
3656 (willem tested upto 60000)[/quote]


Great work, Willem! And a top-5000 prime to boot! :smile: Keep us posted on your progress. I'm accumulating all of the info. for bases <=32 into a master spreadsheet that I am using to slowly create some web pages.

Question...were you able to prove the primes? Although it's slow, I use Proth.



Thanks,
Gary

Siemelink 2007-12-06 20:34

Aah, proving prime with some other program. I didn't think so far ahead yet. Ok, I have proth running now.

Thanks for the tip, laters, Willem.

rogue 2007-12-06 21:52

You should be able to prove these prime with WinPFGW and it will be much faster than Proth. Use the -tp or -tm switch (depending upon -1 or +1, although I don't recall which switch is used with which sign). Combine it with -f0 and you should be all set.

axn 2007-12-07 00:37

[QUOTE=rogue;120072]You should be able to prove these prime with WinPFGW and it will be much faster than Proth. Use the -tp or -tm switch (depending upon -1 or +1, although I don't recall which switch is used with which sign). Combine it with -f0 and you should be all set.[/QUOTE]

-tp means "Classic [B]p[/B]lus side test" = N+1 test = applicable for -1 numbers.
Vice versa for -tm

[quote=pfgw documentation]
-t currently performs a deterministic test. By default this is an N-1
test, but N+1 testing may be selected with '-tp'. N-1 or N+1 is
factored, and Pocklington's or Morrison's Theorem is applied. If 33%
size of N prime factors are available, the Brillhart-Lehmer-Selfridge
test is applied for conclusive proof of primality. If less than 33%
is factored, this test provides 'F-strong' probable primality with
respect to the factored part F.
[/quote]

geoff 2007-12-07 00:52

[QUOTE=gd_barnes;119126]Masser, I checked out the thread that you posted here. It is a good one but I guess I am a little bit confused now. In the post, doesn't Geoff state the following?:

[QUOTE=geoff]
2. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n >= 0.
...
3. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n.
...
For the purposes of [b]this[/b] project it makes no difference whether we use definition 2 or definition 3.
[/QUOTE]

I can only speculate and maybe you can confirm that we want to use the above base 5 defintion for all bases. Is that your thinking? If so, here is where I'm confused:
[/QUOTE]

The conclusion above was based on the particular properties of the base 5 sequences (i.e. the primes that had been discovered so far), it does not a priori apply to other bases.

edit: The quote above has been edited, see the [url=http://www.mersenneforum.org/showthread.php?t=4832]original post[/url].

gd_barnes 2007-12-07 07:52

[quote=geoff;120096]The conclusion above was based on the particular properties of the base 5 sequences (i.e. the primes that had been discovered so far), it does not a priori apply to other bases.

edit: The quote above has been edited, see the [URL="http://www.mersenneforum.org/showthread.php?t=4832"]original post[/URL].[/quote]


Thanks for responding and clarifying Geoff. I'm wondering what properties those were for base 5 and would they apply to at least 'some' other bases such as base 22?


Gary

gd_barnes 2007-12-07 07:55

[quote=rogue;120072]You should be able to prove these prime with WinPFGW and it will be much faster than Proth. Use the -tp or -tm switch (depending upon -1 or +1, although I don't recall which switch is used with which sign). Combine it with -f0 and you should be all set.[/quote]


Very good. I should have known as much since I've been using PFGW extensively for these conjecture searches but didn't look into all of its options. Thanks for the heads up.


Gary

kar_bon 2007-12-07 13:24

something like this:

call:
pfgw -tc -q"1468*11^26258+1"

output:
PFGW Version 20031222.Win_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]

Primality testing 1468*11^26258+1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N-1 BLS with factored part 100.00% and helper 0.02% (300.02% proof)
1468*11^26258+1 is prime! (813.9536s+0.1133s)

karsten

geoff 2007-12-08 02:24

[QUOTE=gd_barnes;120117]Thanks for responding and clarifying Geoff. I'm wondering what properties those were for base 5 and would they apply to at least 'some' other bases such as base 22?[/QUOTE]

Roughly, if the definition of `base B Sierpinski number' is `X is a base B Sierpinski number if X*B^n+1 is never prime for n >= L' then for a given X, once L is reduced below a certain level reducing it further doesn't add any further cases that need to be checked, because X*B^L+1 will not be an integer and therefore not a prime. As more primes are found the level at which L is set can be raised without altering the list of open candidates.

robert44444uk 2007-12-09 12:52

Some notes from unpublished work by Prof Caldwell

Definition of a Sierpinski number

An integer k > 1 is a Sierpinski number base b if gcd(k+1,b-1) = 1 and
k.bn+1 is composite for all n > 0.
• gcd(k+1,b-1) = 1 avoids trivial covers (1-covers).
• k > 1 avoids leading Generalized Fermat divisors. (May use Strong Sierpinski for GFN’s included, and may toss out further GFN’s in the weak case above.)
• n > 0 avoids removing k = p-1 as a multiplier for all primes p and all bases b. (Shouldn’t b be involved in the choice of k?)

gd_barnes 2007-12-10 22:07

[quote=robert44444uk;120272]Some notes from unpublished work by Prof Caldwell

Definition of a Sierpinski number

An integer k > 1 is a Sierpinski number base b if gcd(k+1,b-1) = 1 and
k.bn+1 is composite for all n > 0.
• gcd(k+1,b-1) = 1 avoids trivial covers (1-covers).
• k > 1 avoids leading Generalized Fermat divisors. (May use Strong Sierpinski for GFN’s included, and may toss out further GFN’s in the weak case above.)
• n > 0 avoids removing k = p-1 as a multiplier for all primes p and all bases b. (Shouldn’t b be involved in the choice of k?)[/quote]


Thanks for the info. I understand everything but the last two here. On the 2nd and 3rd ones, can you give specific examples on what you are referring to. Pardon the ignorance. I can understand best by examples. I suspect the 2nd one is related to the issue of 22*22^n+1 and 484*22^n+1 but am not sure.

So by extension on the definition of a Sieprpinski number base b, we can draw the same conclusion on Riesel numbers that have 1-covers by changing k+1 to k-1. That is:

An integer k > 1 is a Riesel number base b if gcd(k-1,b-1) = 1 and
k.bn-1 is composite for all n > 0.
• gcd(k-1,b-1) = 1 avoids trivial covers (1-covers).

In my research and creation of web pages for all conjectures base 2-32, I had already observed this and will put it in 'sentence format' for ease of everyone else's reference (mainly my own) :smile::

A k cannot be a Riesel nor a Sierpinski k if all values of n have a single trival factor. And the elimination of these k's is directly related to the factorization of the base minus 1. Siting specific examples for both types, we have:

Riesel:
Base 2 (none, i.e. there is no factorization of 1)
Base 3 (k==1 mod 2 eliminated with a factor of 2, i.e. the factorization of 2 is simply 2).
Base 4 (k==1 mod 3 eliminated with a factor of 3, i.e. the factorization of 3 is simply 3).
Base 5 (k==1 mod 2 eliminated with a factor of 2, i.e. the factorization of 4 is 2*2)
Base 6 (k==1 mod 5 eliminated with a factor of 5).
Base 7 (k==1 mod 2 eliminated with a factor of 2 AND k==1 mod 3 eliminated with a factor of 3, i.e. the factorization of 6 is 2*3).

Sierpinski:
Bases 2 & 3; same as above
Base 4 (k==2 mod 3 eliminated with a factor of 3).
Base 5; same as above
Base 6; (k==4 mod 5 eliminated with a factor of 5).
Base 7 (k==1 mod 2 eliminated with a factor of 2 AND k==2 mod 3 eliminated with a factor of 3, i.e. the factorization of 6 is 2*3).

And continuing to larger Sierpinski bases we have base 30 where only k==28 mod 29 are gone with a factor of 29 and base 31 where k==1 mod 2 are gone with a factor of 2, k==2 mod 3 are gone with a factor of 3, and k==4 mod 5 are gone with a factor of 5.

Robert, I'm going to send you links to web pages in a PM that I have already created with a lot of information for bases 2-32 for this. I have come up with a most unusual proof for the base 12 Riesel conjecture and I think part of it may be related to the last item that you are referring to here. I will post the proof in the next post here for everyone to review.

You did a lot of coordination on these conjectures and if you could, I'd like to get your feedback on the pages before opening them up to everyone. I also have a reservation page to get the ball rolling again on this.


Thanks,
Gary

gd_barnes 2007-12-10 22:52

I have placed a proof of the Riesel Base 12 conjecture in the 'Sierpinski/Riesel bases 6 to 18' thread.

gd_barnes 2007-12-11 03:51

[quote=kar_bon;120129]something like this:

call:
pfgw -tc -q"1468*11^26258+1"

output:
PFGW Version 20031222.Win_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]

Primality testing 1468*11^26258+1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N-1 BLS with factored part 100.00% and helper 0.02% (300.02% proof)
1468*11^26258+1 is prime! (813.9536s+0.1133s)

karsten[/quote]

Hi Karsten,

Thanks for the feedback. I tried Axn1/Rogue's suggestion of using the -tm switch along with the -f0 switch and it speeded up the search on your prime by nearly 10 times!! Apparently the -f0 switch causes it to not do any trial factoring, which would certainly be unneccesary for a probable prime. -tc apparently does both a +1 and -1 test so it also does more testing than is needed. -tm does just what you need here...it only does a -1 test, which is what is needed for a +1 probable prime. Here is the output:

Primality testing 1468*11^26258+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
1468*11^26258+1 is prime! (83.0808s+0.0065s)

The input to PFGW was: "pfgw -tm -f0 -q1468*11^26258+1"

Particulars of the test:

It was run on a 1.66 Ghz Dell Core duo laptop. (It seems to run about as fast as a 3 Ghz P4.)

My version of PFGW does not accept the quotes around the equation.

Per the README file, I am using PFGW v1.2 Release (January 30, 2005). I downloaded it 2-3 months ago.


I hope this saves you a little time too! :smile: It did me...I needed to test a much larger prime for base 17 that took almost 6 times as long as the one you tested, i.e. 92*17^51311+1, which took 476.1328s+0.0087s.

Thanks Rogue, Axn1, and Karsten for helping!


Gary

gd_barnes 2007-12-17 05:03

Report future status at "Conjectures 'R Us"
 
All base 22 searchers,

All conjectures for bases > 2 except those being worked by other major projects are now being coordinated in the new "Conjectures 'R Us" effort in this Open Projects forum. Please report all future reservations and statuses for base 22 in the reservations/statuses thread for that effort. Web pages have been created that show all current relavent info.

After a couple of days, I'll request that this thread be locked to avoid any duplication of future effort.


Thanks,
Gary


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