Conjectures 'R Us; searches needed

[rogue]

I stopped base 10 at n=195000 for the remaining four k. If someone else wants to continue, they are free to do so.

[Citrix]

For base 31 Sierpinski number is 4562804. (331,19,7,13,37)
One thing that I have noted in generating these numbers is that as the base gets larger the sierpinski/riesel numbers get smaller and smaller and easier to generate.

[gd_barnes]

Originally Posted by rogue

I stopped base 10 at n=195000 for the remaining four k. If someone else wants to continue, they are free to do so.

Thanks for the info. Rogue.

In addition to four more base 10 k's to search, I'm "adding back" the five squared k's for Riesel base 30 that are on the web pages but where I suggested not searching them originally. More analysis convinced me that there is no combintation of numeric and algebraic factors that cover all n so they should all have a prime at some point. All are very low weight though, partly because all even n do have algebraic factors. That is in one of the equations, 25*30^n-1, where n=2*q, it factors to (5*30^q-1)*(5*30^q+1). So if you want to get up to high n-ranges quickly, these will be the ones for you. Or you can just take Rogue's k's. They're already at n=195K.

Rogue's former k's that can now be searched from n=195K:

Riesel's:
4421*10^n-1
7019*10^n-1
8579*10^n-1

Sierpinski:
7666*10^n+1

Riesel base 30 squared k's that can now be searched from n=25K in addition to the k's shown in the first post here:

25*30^n-1
225*30^n-1
1024*30^n-1
1936*30^n-1
2916*30^n-1

IMPORTANT NOTE: The sieving programs will not remove algebraic factors. After sieving to a nominal limit on squared k's such as this, you'll want to manually remove all even n's. Srsieve will show 'Warning: algebraic factors' but I don't know if any of the other sieving programs do. You can use the Excel MOD function, sorting, deleting, and resorting to accomplish this.

Personally, I like to remove them after I've run srsieve to P=500M or 1G but before I feed them to sr2sieve or sr1sieve for deeper sieving.


Gary

[gd_barnes]

Originally Posted by Citrix

For base 31 Sierpinski number is 4562804. (331,19,7,13,37)
One thing that I have noted in generating these numbers is that as the base gets larger the sierpinski/riesel numbers get smaller and smaller and easier to generate.

Oh, EXCELLENT! I had tested it to 5M and for some reason missed it so I'll show it on the page. Robert had indicated the same thing to me about the #'s getting smaller as the bases get larger. Very strange! I would have expected the opposite.

Now, your task for today is to come up with the #'s for bases 3, 7, and 15 and make sure there are no lower ones with numeric covering sets!


Gary

[Citrix]

Originally Posted by gd_barnes

Robert had indicated the same thing to me about the #'s getting smaller as the bases get larger. Very strange! I would have expected the opposite.


Gary

Why is this strange, this should be obvious? Only numbers with b^n-1 =(b-1)*prime, n small will have large Sierpinski and riesel numbers. The interesting question to ask is that can k=2 be a sierpinski or riesel number for any base

[gd_barnes]

Originally Posted by Citrix

Why is this strange, this should be obvious? Only numbers with b^n-1 =(b-1)*prime, n small will have large Sierpinski and riesel numbers. The interesting question to ask is that can k=2 be a sierpinski or riesel number for any base

D'oh! Shall I ask what 2 plus 2 is now? It's funny how when someone puts something in a different light than the way you've been microfocused on it, that it becomes obvious.

That is an interesting question about k=2. Are there any conjectures about it? Have you tested it?

What software did you use to get the conjecture for Sierp Base 31?


G

[gd_barnes]

Originally Posted by Citrix

For base 31 Sierpinski number is 4562804. (331,19,7,13,37)
One thing that I have noted in generating these numbers is that as the base gets larger the sierpinski/riesel numbers get smaller and smaller and easier to generate.

OK, Citrix, I get to throw a D'oh right back at you. Shouldn't it be obvious that 4562804*31^n+1 has a trivial factor of 3. So it's not the conjecture. For base 31, you have to eliminate all k's where k==1 mod 2, 2 mod 3, and 4 mod 5. 4562804 is both == 2 mod 3 and == 4 mod 5.

This is why I had missed it in my testing to k=5M. I eliminated all k's with the above conditions before testing. I'm 99% confident that the conjecture is k>5M. But since the Riesel conjecture is 134718, I wouldn't think it would be very much higher.



Gary

[Citrix]

Originally Posted by gd_barnes

OK, Citrix, I get to throw a D'oh right back at you. Shouldn't it be obvious that 4562804*31^n+1 has a trivial factor of 3. So it's not the conjecture. For base 31, you have to eliminate all k's where k==1 mod 2, 2 mod 3, and 4 mod 5. 4562804 is both == 2 mod 3 and == 4 mod 5.

This is why I had missed it in my testing to k=5M. I eliminated all k's with the above conditions before testing. I'm 99% confident that the conjecture is k>5M. But since the Riesel conjecture is 134718, I wouldn't think it would be very much higher.



Gary

Good one. I did not take the factorization of 31-1 into account. Anyway using the above covering set there are no solutions up to 50 millions. Some other covering set might work, but I have not tested it.

For fixed k and variable base
We know Sierpinski and Riesel numbers for k=4.
None are known for k=3 and k=2.
I have tested these a little bit, but haven't found a candidate. Perhaps no such bases exist.
This is what I was doing some time back (not up to date).
http://www.mersenneforum.org/showthread.php?t=6918

[gd_barnes]

Originally Posted by Citrix

Good one. I did not take the factorization of 31-1 into account. Anyway using the above covering set there are no solutions up to 50 millions. Some other covering set might work, but I have not tested it.

For fixed k and variable base
We know Sierpinski and Riesel numbers for k=4.
None are known for k=3 and k=2.
I have tested these a little bit, but haven't found a candidate. Perhaps no such bases exist.
This is what I was doing some time back (not up to date).
http://www.mersenneforum.org/showthread.php?t=6918

Sounds like another stubborn one like b=3, 7, and 15. No surprise since it's b=2^q-1. I'll have to take it up to 10 million with all covering sets in the near future.

That thread is good and interesting info. Thanks! My intention is to have ALL conjecture type info. in the pages for this effort somewhat like Karsten has done for the RPS effort. It'll be a while but I will put it on my list of things to do to include some or all of the info. in that thread.

Note to all: I have no intention of restricting this effort in any way. If anyone has info. or has done searches on bases > 32, go ahead and forward it my way and I'll eventually get it into the web pages.


Gary

[japelprime]

I think I will try Base 9: 2036*9^n+1 (100K - 200k)

Is that ok or have someone alredy started working on this number?

I have alredy sieved the file to 5,5G with NewPgen and have now started seiving with SR2Sieve.

[robert44444uk]

Originally Posted by Citrix

The interesting question to ask is that can k=2 be a sierpinski or riesel number for any base

The answer is no! k=1,2,3 are not on and it has been proven (but not published). All k exist as non trivial Riesel and Sierpinski for some base except k=2^x-1 to my belief. A few stragglers beyond k=100 are known to be super-hard to find but base values have been discovered that are not trivial, they are just very very large.