Sierpinski/ Riesel bases 6 to 18

[gd_barnes]

Originally Posted by masser

I have no objection at all, but perhaps all of these threads about other bases belong in the new Open Projects sub-forum. As moderator of this project, I feel that we should maintain focus on the base 5 project. The other bases only serve as a distraction to our project.

OK, thanks. Makes sense. I'll get something started over there in the near future. I have a large amount of info. put together about most bases <= 26 but I still have a fair amount of data collection to go from various sources. One of them being the Base 4 thread in Open Projects that Anon referred to.


Gary

[jasong]

If I may make a suggestion:

There are a lot of bases being covered, and I'm not sure how many threads there are going to be. Maybe there should be a reservation thread and a comments thread, with a moderator using the first post as the update area in the reservations thread.

What do you think?

Edit: and now I see it's already there. Sorry for clogging the thread.

[gd_barnes]

Originally Posted by jasong

If I may make a suggestion:

There are a lot of bases being covered, and I'm not sure how many threads there are going to be. Maybe there should be a reservation thread and a comments thread, with a moderator using the first post as the update area in the reservations thread.

What do you think?

Edit: and now I see it's already there. Sorry for clogging the thread.

Sometime in the next 1-2 weeks, I will have a huge amount of information about all bases <= 32 put into 2-3 web pages and will start an entirely new thread here in 'open projects'. I'm still pulling info. from various sources and 'filling in some holes' on a few bases by doing some testing here and there. After doing a lot of additional searching myself, I've proven several bases not previously proven in this thread and gotten it down to where there are currently 2 Riesel and 6 Serpinski bases <= 32 that have only ONE remaining stubborn k that needs a prime. After additional searching, I'll be asking for help with those one-k bases remaining, on some bases that have many remaining candidates that need a prime, and on starting some bases that have had no work done yet.

I'll put in some info. on one of the pages about reservations.

The Riesel and Serpinski conjectures make for most interesting Prime Search projects in all bases and bringing all of the info. together for multiple bases has both been very interesting and a great learning experience!


Gary

[gd_barnes]

Here is a proof of the Riesel conjecture base 12 that was previously analyzed and searched in this thread:

Conjecture: The Riesel k = 376 with a covering set of 5, 13, 29. This was already given here.

k's where k==1 mod 11 are eliminated with a trivial factor of 11.

There are 3 k's remaining where a prime has not been found. They are k=25, 27, and 64.

Before proving this, for everyone's "entertainment" , here are the top-10 k's with the largest first primes found:
k / (n)
157 (285)
46 (194)
259 (40)
304 (40)
94 (36)
292 (30)
147 (28)
301 (27)
349 (25)
58 (23)


"My conjecture": k=25, 27, and 64 are composite for all n but do not have a specific covering set of numeric factors.

Proof of "My conjecture" for k=25 and 64 generalized for all possible values of k:

For all k=m^2 and k==12 mod 13 (both must be true), the following algebraic and numeric factors are present for all n:
1. For all odd n, there is a factor of 13.
2. For all even n, let k=m^2 and let n=2*q. There are now algebraic factors of (m*12^q - 1) * (m*12^q + 1).

Therefore 25*12^n-1 and 64*12^n-1 must be composite for all n.


Proof of "My conjecture" for k=27 generalized for all possible values of k:

For all k=3*m^2 and k==1 mod 13 (both must be true), the following algebraic and numeric factors are present for all n:
1. For all even n, there is a factor of 13.
2. For all odd n, let k=3*m^2 and let n=2*q-1. There are now algebraic factors of [m*3^q*2^(2q-1) - 1] * [m*3^q*2^(2q-1) + 1].

Therefore 27*12^n-1 must be composite for all n.


This of course begs the question: What really is the Riesel number base 12? IMHO, it is still k=376 (not k=25) and the above examples are the equivalent of trivial k's base 12, i.e. k=1, 12, 23, 34, etc., and just needed to be proven as such. But I'm open to hearing anything on the matter.

It also begs the question of more specifically defining a covering set. I think I've demonstrated here that a k can have both a partial covering set of 'numeric' factors as well as a partial covering set of algebraic factors such that it becomes a full covering set. But once again, IMHO I would think we'd want only "numeric" (not algebraic) covering sets for the conjectures otherwise we'd wind up with very low k's for the conjectures for many bases such as this one.

Any other thoughts, comments, corrections, and opinions are also welcome.


Gary

[gd_barnes]

Originally Posted by jasong

Edit: By the way guys:

2857*16^5478+1 is prime
2158*16^10906+1 is prime
4885 tested to n=50,000(base-16) no primes
3061 tested to n=50000(base-16) no primes
2908 tested to n=38000(base-16) no primes

I'm unreserving all my numbers.

Thanks.

Jasong,

I'm doing some double-checking before posting my web pages...

Neither one of these numbers is prime.

2857*16^5478+1 has a factor of 11. It looks like you may have tested base 2 because 2857*2^5478+1 is prime. But since n is not a multiple of 4, it doesn't help base 16.

2158*16^10906+1 also has a factor of 11. 2158*2^10906+1 has a factor of 127. So I'm not sure what you tested there.

I show both k's as composite to n=6580 and will be testing all k's above n=10K for base 16 before publishing the pages.

If you have a prime on one of these k's, please let me know.


Thanks,
Gary

[jasong]

Originally Posted by gd_barnes

Jasong,

I'm doing some double-checking before posting my web pages...

Neither one of these numbers is prime.

Thanks for the double check, I have no idea what went wrong.

[gd_barnes]

Originally Posted by jasong

Thanks for the double check, I have no idea what went wrong.

No problem...I found the primes after a little more searching:

2857*16^6832+1 is prime
2158*16^10905+1 is prime
And one more from your other k's:
3061*16^8322+1 is prime


All k's below the Sierpinski k=66741 for base 16 are now searched to n=13K. 77 k's are remaining after eliminating all k's with higher primes found by prior projects on bases 2 and 4...not too bad for such a low search range. (3 of the 77 are in effect still being searched by those projects.)

My web pages with most known Riesel/Sierpinski conjecture info. bases 2-32 are complete. Check for a new thread here on Thursday. I'll have some sieved files for several different bases ready to be handed out and searched.


Gary

[gd_barnes]

All bases 6 to 18 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 bases 6 to 18 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