Riesel bases 560 and 758
 

Primes found:

2*560^36-1
3*560^6-1
4*560^1-1
5*560^2-1
6*560^1-1
7*560^1-1
8*560^19904-1
9*560^1-1

2*758^4-1
3*758^1-1
4*758^15573-1
5*758^6-1
6*758^1-1
7*758^67-1
8*758^14-1
9*758^13-1

With a conjectured k of 10, both of these are proven.

Riesel bases 527, 548, and 812
 

Primes found:

2*527^24-1
6*527^42-1
8*527^14-1

Conjectured k = 10. k = 4 remains.

2*548^4-1
3*548^14-1
4*548^45-1
5*548^8-1
6*548^2-1
8*548^2-1
9*548^1-1
10*548^1-1
11*548^2-1
12*548^14-1

Conjectured k = 13. k = 7 remains.

2*812^10-1
3*812^3-1
5*812^50-1
6*812^1-1
7*812^1-1
8*812^8-1
9*812^1-1
10*812^1575-1
11*812^2-1
12*812^1-1

Conjectured k = 13. k = 4 remains.

All have been tested to n=25000 and have been released. As far as I can tell there are no complete algebraic factorizations for the remaining k on these conjectures.

Yes, this is more than two for today, but this provides results for the remaining Riesel conjectures with k <= 13.

New bases S650 and S797 k=8 conjectures are complete to n=25K.

Only k=4 remains on both of them.

This completes all k<=8 conjectures on both sides to n=25K.

Riesel base 863, k=14
Primes:
2*863^4-1
6*863^2-1
10*863^1-1
12*863^3-1
4*863^2403-1

k=8 proven composite by partial algebraic factors
Base proven.

Riesel base 577, k=18
Primes:
2*577^1-1
6*577^1-1
8*577^2-1
12*577^17-1
14*577^5775-1

Trivially factors: k=4,10,16
Base proven.

[quote=gd_barnes;211787]New bases S650 and S797 k=8 conjectures are complete to n=25K.

Only k=4 remains on both of them.

This completes all k=8 conjectures on both sides to n=25K.[/quote]
Hmm...interesting how just k=4 remains on quite a few of these k=8 conjectures. Is there something special about k=4 that makes it extra stubborn?

[quote=mdettweiler;211791]Hmm...interesting how just k=4 remains on quite a few of these k=8 conjectures. Is there something special about k=4 that makes it extra stubborn?[/quote]

It's 4 times a 4th power, which elimates all n's divisible by 4 on all bases and hence makes them somewhat lower weight. But other than that, no, none that I can tell. Using that logic, k's that are perfect squares on the Riesel side should be much worse since their n's cannot be divisible by 2. But my perception is that Sierp k=4 is worse than Riesel perfect squares and I don't have an explanation of why.

One thing that I did recently is see how many bases <= 1024 have k=4 remaining at n=5K. There were 43 of them. Compare that to the following # of bases remaining at n=5K:

Riesel k=2 25
Sierp k=2 35
Riesel k=4 30
Sierp k=4 43

Riesel k=4 was helped somewhat by having some bases k=4 eliminated due to partial algebraic factors making a full covering set but not that much difference.

The Sierp side is definitely tougher for k=2 and k=4, especially on small-conjectured bases.

Explantion of the elimination of n==(0 mod 4) for Sierp k=4:

4b^4 + 1 = (2b^2+2b+1) * (2b^2-2b+1)

In all cases that I looked at for b<=1024 and k=4, this does not make a full covering set so the searches must continue. Where it does make a full covering set is on bases 55 and 81 for k=2500, which is k=4*5^4. Hence you'll see on the pages that those k's are eliminated.


Gary

[quote=rogue;211727]Primes found:

2*548^4-1
3*548^14-1
4*548^45-1
5*548^8-1
6*548^2-1
8*548^2-1
9*548^1-1
10*548^1-1
11*548^2-1
12*548^14-1

Conjectured k = 13. k = 7 remains.

2*812^10-1
3*812^3-1
5*812^50-1
6*812^1-1
7*812^1-1
8*812^8-1
9*812^1-1
10*812^1575-1
11*812^2-1
12*812^1-1

Conjectured k = 13. k = 4 remains.
[/quote]


Well, you ended up with only 3 new bases for the day instead of 5. (hooray!) :-) Riesel bases 548 and 812 had already been done. See:

[URL]http://www.mersenneforum.org/showpost.php?p=209562&postcount=306[/URL]
[URL]http://www.mersenneforum.org/showpost.php?p=209597&postcount=308[/URL]

I see that the untested Riesel thread may have thrown you off there because I still had those 2 as untested. I would suggest double-checking it against the pages before starting a search. The pages should always be within ~2-3 days of up to date. I do my best to keep up with the untested thread but with it sorted by CK, if I forget removing something, there is not an easy way for me to double check myself.


Gary

[quote=unconnected;211788]Riesel base 863, k=14
Primes:
2*863^4-1
6*863^2-1
10*863^1-1
12*863^3-1
4*863^2403-1

k=8 proven composite by partial algebraic factors
Base proven.[/quote]

How is k=8 proven composite by partial algebraic factors?

8*863^4492-1 is prime!

Short analysis:
n==(1 mod 2); factor of 3
n==(0 mod 3); algebraic factors because a^3*b^3-1 has a factor of a*b-1

This leaves n==(2 or 4 mod 6) that need to be searched.

The best example for a small n is n=16, which has a 15-digit smallest factor, i.e.:
290,080,942,920,023 *
2,610,619,153,408,518,748,349,564,802,570,449

Now the base is proven. :-)


Gary

[QUOTE=gd_barnes;211847]I see that the untested Riesel thread may have thrown you off there because I still had those 2 as untested. I would suggest double-checking it against the pages before starting a search. The pages should always be within ~2-3 days of up to date. I do my best to keep up with the untested thread but with it sorted by CK, if I forget removing something, there is not an easy way for me to double check myself.[/QUOTE]

Typically I do double-check, but I only check the last page in the forum, thinking that previous pages would have posts that you have already handled. In this case I bet that I didn't go back to previous pages to verify that nobody else had worked on them. I'll have to remember that next time.

Riesel base 666, k=898
Remaining k's:
74*666^n-1
139*666^n-1

k=144 and k=289 proven composite by partial algebraic factors (even n - diff. of squares, odd n - factor of 29).
Trivially factors - 316 k's.

Primes attached.