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Riesel Sierpinski in base 5

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Robert     Message 1 of 6  Sep 17, 2004
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I apologise in advance if what is shown below is common knowledge or 
discovered previously, but I can find nothing on the web.

The interest shown in this group in Sierpinski/ Riesel base 3 got me 
thinking about base 5. There is a good reason for this. There is a 
small covering set, which repeats every 12n. 

This set is 3,7,13,31,601 does not provide a fairly low value for k 
for the Sierpinski set, in fact, unless my maths (or my typing) is 
wrong, the first value is 9825724, which is larger than the multiple 
3.7.13.31.601 because the lower value 4739461 is not divisible by 2. 
Certainly this value is a Sierpinski number base 5. It may not be 
the smallest though.

Another covering set which might produce a result is 
3,13,17,31,313,11489 which repeats every 80n, but I have not done 
this calculation yet. At the 60 level, I can't seem to come up with 
a covering set with less than 9 values for some reason, but my 
program is not very clever - it is a blackboard equivalent known as 
Excel. Please don't laugh.   

However on the Riesel side, the result is the very satisfying 
346802, using the covering set 3,7,13,31,601. That means there are 
only 173400 values of k to check to see if this is the lowest k 
without primes or probable primes in the power series k.5^n-1

So far, I have checked almost up to n=10000 and have managed to 
eliminate all but about 600 values, (finding primes or probable 
primes 3-prp for the others) and some of these are going to be hard 
to find given that the 5 power series get larger than the 2 power 
series alarmingly quickly.  I eliminated many numbers with a simple 
program written for Maple taking n up to 1000 and with WinPFGW after 
that. 

I suppose it is possible to use the dual approach, but there are 
purists out there I know who think this is unsatisfactory (I 
personally don't). 

I have tested about 300 of these values to see if there are other 
covering sets, but found none up to around k=186000, testing with 
prime divisors up to 500000, using NewPgen. Based on this, I would 
conjecture that 346802 is the lowest n for the Riesel Base 5 
question. 

If anyone is interested in tackling a few of the remaining 600 or so 
values of k I could post them to this group. If I do so, I would 
recommend that people take 20 or so at a time, show what they have 
reserved, take them up to an agreed value of n and then post the 
results to this site. 

Regards

Robert Smith
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Guido Smetrijns     Message 2 of 6  Sep 17, 2004
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Hi Robert,

I think you are right about the smallest Riesel number (346802) but for the
plus-case I did a quick test and came up with the (even smaller)
Sierpinski-number 159986, with the same covering set.
Anyone better?

Kind regards,
Guido.

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Robert     Message 3 of 6  Sep 17, 2004
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Nice find! I realise that my Excel methodology is truly primitive. 
That makes the Sierpinski set even more interesting!

Do you know if this is the smallest value with that covering set?

Regards

Robert Smith


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Guido Smetrijns     Message 4 of 6  Sep 17, 2004
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I really think 159986 is indeed the smallest Sierpinski number, at least
with {3,7,13,31,601} as covering set.

Actually, I only calculated the Riesel numbers and, to be more complete,
given that P=2*3*7*13*31*601=10172526,
I think ALL Riesel numbers with that covering set are given by the following
list (n=0,1,2,3,...) :
{   346802+Pn,
1284362+Pn,
1591472+Pn,
1734010+Pn,
2002508+Pn,
2660146+Pn,
3128204+Pn,
4469512+Pn,
5096456+Pn,
5137228+Pn,
5341088+Pn,
5468494+Pn,
5656006+Pn,
6172876+Pn,
6360388+Pn,
6421810+Pn,
6997418+Pn,
7934978+Pn,
7957360+Pn,
8670050+Pn,
9157312+Pn,
9269222+Pn,
9372596+Pn,
10012540+Pn}.

From this list all Sierpinski numbers can easily be calculated by the
formula s=P-r,
so the smallest one is given by 10172526-10012540=159986.

I didn't check any other covering sets, though...

Cheers,
Guido.

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Mikael Klasson     Message 5 of 6  Sep 17, 2004
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Hi,

> Nice find! I realise that my Excel methodology is truly primitive. 
> That makes the Sierpinski set even more interesting!
> 
> Do you know if this is the smallest value with that covering set?

It is. No smaller k exists for either the + or - side, using a covering 
set with only primes p less than 10000 and order(5,p)<=1000.

Mikael
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Hans.Rosenthal@t-online.de     Message 6 of 6  Sep 18, 2004
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Robert,

Do never have a poor opinion of something that you reached with your
own method. Even, or especially, if you used tools like Excel or Word.

Sometimes (in fact, most of the times) it's enough to only use WordPad
for preparing a find of some interest. As an example, I only used the
very old Windows 95 version of WordPad to prepare the batch files that
helped to find a prime gap of 1310 (with merit 25.7495) following the
23-digit prime 12433743091180061144441 which would easily fit into the
Nicely's collection of prime gaps.

(You will not yet find it there, since this is only the beginning...)

Hans

Robert Smith wrote:

 > Nice find! I realise that my Excel methodology is truly primitive.
> That makes the Sierpinski set even more interesting!
> 
> Do you know if this is the smallest value with that covering set?
> 
> Regards
> 
> Robert Smith 
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