Group Effort

[robert44444uk]

Chaps, I understand that there is some desire to ensure that top 500o primes get reported as a "group" effort, i.e. your name with attribution to a separate counting prime provider called "Sierpinski Base 5"

Hwo do people think about this? Shall we vote?

Regards

Robert Smith

[geoff]

Another member suggested that project codes be created and I said go ahead, I should have asked if there were any objections first, sorry.

Each prime discovered is pretty much an individual effort, since everyone is doing their own sieving.

But I think it would be good to have a project code so that when someone sees one of the primes in the top 5000 database they can see that it is not just a number someone pulled from their hat, but that it serves a purpose.

We could leave it up to each individual discoverer whether or not they want to add the project code to the list of credits for their primes, at least while there is no central sieving effort going on.

[ValerieVonck]

Update:

When I get confirmation from Chris Calldwell, I will post the links to the project pages here.

Things already done:
- Created 2 projects on the prime pages: 1 for Riesel Base 5 & 1 for Sierpinski Base 5
- Mailed the links for the top 5000 primes to Chris

To Do:
- Chris has to contact the discoverers of these primes so that they (or he) can add them to these projects.

Regards,
Cedric

[robert44444uk]

Ok, I certainly don't have any objections. So lets move forward on that basis.

Regards

Robert Smith

[ValerieVonck]

Originally Posted by robert44444uk

Ok, I certainly don't have any objections. So lets move forward on that basis.

Regards

Robert Smith

Thnx!!!

[ValerieVonck]

Still no update from Chris Calldwell..
If I don't hear from him tomorrow, I will post the links here.
Stay tuned!

[ValerieVonck]

Still no update from Chris.

So I will post the links to the projects here:

http://primes.utm.edu/bios/page.php?id=768 (Riesel Base 5)

http://primes.utm.edu/bios/page.php?id=769 (Sierpinski Base 5)

[geoff]

Hi Cedric, thanks for creating these project pages. Could you change the descriptions to something like:

This project is searching for a prime of the form k*5^n+1 for each even k less than 159986, to prove that 159986 is the smallest even base five Sierpinski number.

and

This project is searching for a prime of the form k*5^n-1 for each even k less than 346802, to prove that 346802 is the smallest even base five Riesel number.


BTW, does anyone have a reference to the proof that 159986 / 346802 are actually base 5 Sierpinski / Riesel numbers?

[ValerieVonck]

Geoff,

I will change the descriptions asap.

Cedric

[robert44444uk]

Geoff asks:

BTW, does anyone have a reference to the proof that 159986 / 346802 are actually base 5 Sierpinski / Riesel numbers?

Just run the numbers through a sieve such as NewPGen, it eliminates all n immediately.

I found these numbers out by trying to find a simple covering set, and this is when I discovered that these sets exist, it is a question of finding the smallest k which gives the cover in this set.

For example look at various small primes with low order base 5, and you soon determine that you can design k such that these primes would cover all of a group of n. I am not at home so I cannot tell you offhand which is the size of group of n which allows cover with small primes. These groups tend to be numbers which very large numbers of factors, such as 12,24,60,120 and these factors correspond to the order base 5 of the small primes. For example, a range of 12 n can be eliminated on a recurring basis in a number of ways.

For example. by one prime with order base 5 of 2, one prime of order base 5 of 3, one prime of order base 5 of 4, and two of order base 5 of either 6 or 12

1 X....X
2 ..........X
3 X
4 .....X
5 X
6 .........X........X
7 X...X
8 ..............X
9 X
10 .....X...X
11 X
12 ..................X

I will find out the actual covering set for Sierpinski 5 the next time I am home long enough to dig out my workings and supply this to the group.

The problem is that maybe there are other covering sets which happen to provide cover for a smaller k, and that is what this search is all about. It is, in my opinion highly unlikely that there is a smaller k, but as we have seen from Sierpinski base 2 (SoB project) , it takes a lot of effort to show this is the case, because you have to find a prime to disprove.

Regards

Robert Smith

[robert44444uk]

The prime 3 is 2 order base 5
Similarly 31 is 3 order base 5
13 is 4 base 5
7 is 6 order base 5
601 is 12 order base 5

The table I looked at was:

1 X
2 .......X
3 X
4 .................X
5 X.....X
6 .........................X
7 X
8 .......X........X
9 X
10 .................................X (601)
11 X....X
12 ...............X.......X

Lets look at 601, only 1/601 k's will ensure 601 appears as a factor in the power series k*2^5-1, and in the instance I gave, at the case k*5^10-1, this means that the k must be 25mod601 as mod(25*5^10-1, 601)=0

So the smallest k which is Riesel is 25mod601, or (a multiple of 601)+25.

You can show (in my case, by trial and error) that 577 is the smallest number which provides the right mods for the other four primes.

And then 577*601+25=346802

Proving this to be the smallest is our task. I have had it confirmed that no other k provides for a smaller covering set using small primes.

We can, perhaps, see this if we look at the other primes with small order base 5. These are:

11 and 71 are 5 order base 5
19531 is 7 order base 5
313 is 8 order base 5
829 is 9 order base 5
521 is 10 order base 5
12207031 is 11 order base 5

And that is the whole list of primes with order <=12. Their orders base 5 are not factors of 12, so that any covering set would have to look at a larger range of n.

The only smaller range of n might be 10. For example to include 11 and 71, we would need a range of n of a multiple of 5. If we look at 10, then we have 4 possible positions covered with 11 and 71, and a further 3 could be covered with the prime 3 and one more with 521. However, that is only 8 positions out of 10 covered with no further possibilities of cover with small primes.

In general it is only worth looking at any set of primes where the conditions 1) a set of primes exists with orders base 5 which are all factors of any given composite and where 2) for that set, Sigma(1/order base 5) is >= than 1, are both satisfied. Even then, therre is guarantee that you can line them up to eliminate every possible combination. In the example just quoted, Sigma (1/order base 5) is 1/5+1/5+1/10+1/2 = 1 exactly, but only 80% of positions can be covered.

As you can see my maths is shaky. This is no proof, but it shows my thinking when I arrived at the Riesel number that I think is the smallest. I was lucky because my sequential combination of the 5 primes happened to give this number. For this reason I was not able to find the lowest k for the Sierpinski series. I hypothesised a higher k and someone else found a different sequencing of the five primes which make up the covering set, which if I remember were 3,13,7,313 and 601, all of whose order base 5 are again factors of 12.

Hope this helps. If someone wants to lay this out in mathematical notation they are very welcome.

Regards

Robert Smith