Riesel/Sierp #'s for bases 3, 7, and 15

[Siemelink]

Hidiho,

I've done some programming this week and this is what I found:

Base 7 cover set = 5, 13, 19, 43, 73, 181, 193, 1201
Smallest Riesel = 408034255082

Base 15 cover set = 13, 17, 113, 211, 241, 1489, 3877
Smallest Riesel = 36370321851498

I'll be tinkering a bit more with my code and then I'll show it here on the forum.

Laters, Willem.

[gd_barnes]

Originally Posted by Siemelink

Hidiho,

I've done some programming this week and this is what I found:

Base 7 cover set = 5, 13, 19, 43, 73, 181, 193, 1201
Smallest Riesel = 408034255082

Base 15 cover set = 13, 17, 113, 211, 241, 1489, 3877
Smallest Riesel = 36370321851498

I'll be tinkering a bit more with my code and then I'll show it here on the forum.

Laters, Willem.

If you are correct, this is HUGE, especially for base 7 where the conjecture dropped substantially! I'll do some verification myself and if they are correct, I will change the web pages.

Can you do the same thing for the Sierp side on both bases?


Gary

[gd_barnes]

I've now confirmed these to be correct, although cannot guarantee that they are the lowest Riesel values. Proofs:

408034255082*7^n-1:

Factor   n-occurrences   n-remaining
19       n==(1 mod 3)    n==(0,2 mod 3)
5        n==(3 mod 4)    n==(0,2,5,6,8,9 mod 12)
43       n==(2 mod 6)    n==(0,5,6,9 mod 12)
1201     n==(1 mod 8)    n==(0,5,6,12,18,21 mod 24)
13       n==(6 mod 12)   n==(0,5,12,21 mod 24)
181      n==(0 mod 12)   n==(5,21 mod 24)
73       n==(5 mod 24)   n==(21 mod 24)
193      n==(21 mod 24)  (none)

36370321851498*15^n-1:

Factor   n-occurrences   n-remaining
241      n==(1 mod 3)    n==(0,2 mod 3)
113      n==(2 mod 4)    n==(0,3,5,8,9,11 mod 12)
211      n==(3 mod 6)    n==(0,5,8,11 mod 12)
17       n==(4 mod 8)    n==(0,5,8,11,17,23 mod 24)
1489     n==(0 mod 8)    n==(5,11 mod 12)
13       n==(5 mod 12)   n==(11 mod 12)
3877     n==(11 mod 12)  (none)

A nice piece of programming Willem!

The Riesel conjecture web pages have now been updated.


Gary

[Siemelink]

They are the lowest for this cover set.
There may be different cover sets that repeat every 24n. But those also do not give a lower riesel.

I will check a bit deeper (36n or 48n) but my program isn't ready for that yet.
I need to improve on the efficiency before I can tackle base 3. The proposed cover set for that one repeats every 144n.

Laters, Willem.

[robert44444uk]

Good work Siemelink. The real challenge is in base 3, where I would like to think there is a really much lower Sierpinski and Riesel.

[masser]

Wow! That is really great work. Having studied these ideas in the past, I always appreciate seeing someone find lower Riesel and Sierpinski k's and the corresponding covering sets.

Congrats.

P.S. You may be able to "brute force" the base 7 result now. This can be done with looping in NewPGen and/or pfgw, I believe.

[R. Gerbicz]

Originally Posted by gd_barnes

I've now confirmed these to be correct, although cannot guarantee that they are the lowest Riesel values.

1*7^n-1 is also composite for every positive n, so k=1 would be the smallest Riesel value, or what are you searching? If you accept only even k values, then 4*7^n-1 is also composite for every positive n, because it's >3 and divisible by 3.

[gd_barnes]

Originally Posted by R. Gerbicz

1*7^n-1 is also composite for every positive n, so k=1 would be the smallest Riesel value, or what are you searching? If you accept only even k values, then 4*7^n-1 is also composite for every positive n, because it's >3 and divisible by 3.

k=1 and k=4 have trivial factors of 3 for all n-values and hence are not considered. For Riesel base 7, we do not consider k==(1 mod 3) where all n-values have a trivial factor of 3 nor k==(1 mod 2) where all n-values have a trivial factor of 2.

Therefore for Riesel base 7, we only consider k==(0 mod 6) and k==(2 mod 6). Taking it further, for Sierp base 7, we would only consider k==(0 mod 6) and k==(4 mod 6).


Gary

[Siemelink]

Smallest Riesel for base 3 = 1910197852104712
Cover set = {5, 7, 13, 17, 41, 73, 97, 193, 577, 6481}
With factor in sequence length 48:
5 6481 13 7 5 13 41 73 13 7 17 13 5 97 13 7 5 13 193 73 13 7 41 13 5 6481 13 7 5 13 41 73 13 7 193 13 5 577 13 7 5 13 17 73 13 7 41 13

[Siemelink]

Smallest Riesel for base 3 = 1200424637252
Cover set = {5, 7, 13, 19, 37, 41, 73, 757, 6481}
With factor in sequence length 72:
13 19 6481 13 5 7 13 37 5 13 73 7 13 757 41 13 5 7 13 19 5 13 41 7 13 37 6481 13 5 7 13 757 5 13 73 7 13 19 41 13 5 7 13 37 5 13 41 7 13 757 6481 13 5 7 13 19 5 13 757 7 13 37 41 13 5 7 13 757 5 13 41 7

[masser]

The notation is slightly different on the linked webpage, but the point is that

2*31532322469*3^n-1 is always composite.


http://tech.groups.yahoo.com/group/p...m/message/4698