- Jan 5, 2007I thought I would post some results of my limited studies of

Sierpinski/ Riesel series in bases other than 2,3,4 and 5. Some of you

with efficient programming should be able to take this study further.

Here in Bangladesh I am unable to access my Maple software, which

would have speeded things up a lot.

Some of you might have seen recent Yahoo postings relating to the

hypothesis that there is a covering set for every base for both Riesel

and Sierpinski series. This is going to be difficult to prove as such,

but certainly it seems very likely. I am likely to move my research to

the special base form base=2^n-1, where I need to find a covering set

in principle for all values and these may not be easy to find.

Anyway here is information on bases 6 to 18:

Base 6:

Will need a lot of work a covering set, repeating every 12 n is

[7,13,31,37,43]. Lowest Sierpinski and Reisel candidates not known.

But neither can be greater than 4488211.

Another possible covering set, with repeat of 12 is [7,43,37,31,97]

Base 7:

Totally horrible. Possible covering set with repeat every 24 n is

[19,5,43,1201,13,181,193,73], also 5 other sets perming 73, 193 and 409.

Sierpinski and Riesel numbers are both lower than 162643669672445

Work is needed to find a low k value which is Riesel or Sierpinski.

Base 8:

Covering set [3,5,13] covering every 4 n. The corresponding Sierpinski

number is 47, but it is not proven for the small fact that k=1 is

known not to have small primes. (Think about it: 8^n+1= 2^3n+1

For Reisel k=112 looks the most likely candidate. A prime needs to be

found however for k=14

Base 9:

Covering set every 6n for [5,7,13,73]. Alternative covering set every

8 n for [5,41,17,193]. Lowest mooted Sierpinski is 2344 (k=439 is not

Sierpinski because the k is also trivial). Lowest conjectured Riesel

is 74, so should be easy to prove, but 4,16,36,64 are proving pesky.

Note 16 and 64 are subsets of 4.

Base 10:

Covering set every 6n is [11,37,7,13]. Lowest known non trivial

Sierpinski is 9351, lowest Riesel 10176. No work done yet to prove these.

Base 11:

Covering set every 6n is [3,7,19,37]. Lowest Sierpinski k is thought

to be 1490. 3 ks need to be eliminated 416, 958, 1468

Lowest Riesel is thought to be 4624. 12k's still need to be

eliminated. 62, 682, 862, 904, 1528, 2410, 2690, 3110, 3544, 3788,

4208, 4564.

Base 12:

Covering set every 6k is [13,157,7,19]. Lowest Sierpinski mooted to be

14600 and lowest Riesel 16329. No work on this yet.

Base 13:

Covering set every 3 n, [7,17,5]. Lowest Sierpinski is proven to be

132. Lowest non trivial Riesel is thought to be 302, but need to prove

k=288 is not.

Base 14:

With a covering set every 2n [3,5] this proves to be easy. Proven

Sierpinski and Riesel values are both proven at 4.

Base 15:

Horrible. A covering set is [241,113,211,17,1489,13,3877], and

Sierpinski and Riesel values are therefore less than 7330957703181619.

As bad as the base 3 problem.

Base 16:

Sierpinski number not known, quite complex to calculate. Covering set

[7,13,19,37,73] have 36 combinations to check Sadly the simple

covering set 7,13,17 only appears in trivial solutions. On the Riesel

side not so sad, conjectured to be 120. However n=9 is a problem, this

number is well studied as 9.16^n-1 = 9.2^(4n)-1, and no values for n

less than 207177

Base 17:

Covering set every 4 n, [3,5,29]. Sierpinski value of 278, the

following candidates need a prime 88,92,160,244,262. On the Riesel

side, the lowest is conjectured to be 86, with only k=44 needing to

find a prime.

Base 18:

Covering set every 6n is [19,13,5]. Sierpinski value is 398. 4 k

candidates seek primes 18,324,122,381. Riesel is proven to be 246

Any help anyone can give in taking this to higher bases, or proving

the mooted Sierpinskis and Riesels is welcome. Postings please to

http://www.mersenneforum.org/showthread.php?t=6895

Regards

Robert Smith