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 robert44444uk 2006-02-04 12:21

Largest Simultaneous Primes

I have been in correspondence with Jens Kruse Andersen, who maintains the records for simultaneous primes, with a view of accepting the Octoproth forms into the record books.

His response, quoted with his permission:

"There are several things I would like before allowing your forms (less might
do in some cases):

A set of simultaneous primes should be uniquely determined from one of the
primes and its position in the set.
I think there might be different octoproth (n,k) pairs with the same starting
prime 2^n-k.
I think the limitation k<2^(n-1) would prevent such situations in a reasonable
way.
Am I right?
I am not asking you to change your definition, only to accept a limitation in
which sets can be listed at my site.

[url]http://hjem.get2net.dk/jka/math/simultprime.htm[/url] names the type in the tables.
A common 4?/8/12/16/? name, e.g. "multiproth", would be nice for reference.
Have you considered 4-sets (2^n+-k, k*2^n+-1) ?

The currently allowed forms have a maintained record page which is linked from
my page.
Such a multiproth page would be good for easy comparison.
It only needs to keep the single largest case for each number of primes.

The nth of 2n primes in a set counts for size in comparison to other
constellation types. Multiproths get a disadvantage because the largest n
primes are considerably larger. You must accept this handicap. I don't want to
credit a size where almost all prp tests in the search were on significantly
smaller numbers."

So I think we have a chance here to qualify as we restrict the 2^n-k form.

I cannot get onto his site at present - I am timed out - to find out what the records are for 8- 12- and 16- and we are at a disadvantage as mentioned above, but maybe we should be trying for these records? What do people think?

Regards

Robert Smith

 Kosmaj 2006-02-04 14:33

After several attempts (over half an hour) I was able to open his page. The current records are (using i-th prime for our [I]multi[/I]-Proths):

8-primes: 206 digits (tuplet), Octoproth (n=346): 105 digits
12-primes: 44 digits (bitwin), Dodecaproth (n=91): 28 digits
16-primes: 26 digits (tuplet), Hexadecaproth (if we find one for n=76): 24 digits

The hendicap of using the i-th prime (among 2i primes) is not a big problem for Octoproths but it is for Hexadecaproths. For example, let's suppose we find a Hexadecaproth for n=76 and a 17-digit [I]k[/I]. Then, we'll have 2 primes with 23 digits, 6 primes with 24 digits and 8 primes with 39-41 digits which I think (obviously?) is better than 16 primes with 26 digits, but based on 8-th prime with 24 digits our Hexadecaproth will be ranked below the 26-digit tuplet.

Another possible size measure I was thinking about is a "score" used by prof. Caldwell on Top-5000 (or a variation of it) summed over all primes in the group.

In any case approaching existing record in any catogery (4, 8, 12) will require a huge computation effort which presently I'm not ready to undertake.

Finally, I don't have a problem accepting his second condition, k<2^(n-1), since we begin the search from k=1, not from k=2^n-1, and thus all our large multi-Proths satisfy the condition.

 robert44444uk 2006-02-04 14:34

Records to date

Based on what I have seen on the largest simultaneous primes site, we will need an octoproth with 207 digits, which equates to working on 2^688 or above.

For dodecaproths we need to get 45 digits, equivalent to 2^147 or better

Regards

Robert Smith

 R. Gerbicz 2006-02-04 15:56

[QUOTE=Kosmaj]
For example, let's suppose we find a Hexadecaproth for n=76 and a 17-digit [I]k[/I]. Then, we'll have 2 primes with 23 digits, 6 primes with 24 digits and 8 primes with 39-41 digits which I think (obviously?) is better than 16 primes with 26 digits, but based on 8-th prime with 24 digits our Hexadecaproth will be ranked below the 26-digit tuplet.
[/QUOTE]
For n=76 the smallest hexadecaproth'k k value probably will have got 22 digits. 17 digits is very low! For example you've completed the dodecaproth search for n=76 up to 60000T, and 60000T has got exactly 17 digits...

Another note: Now I'm writing a program to search 22 primes in arithmetic progression. The world record is 23 primes.

 robert44444uk 2006-02-04 18:45

Programs

Actually Robert I am sure your current program for Octos could be adapted quite easily to search for long chains of bitwins, (for which the longest is currently 7). If you were interested in optimising this, I would be interested as I am still trying to find the k which will provide the first instance of 10 twins, ten CC1 and ten CC2 in the series k.2^n+/-1. Bitwins of length 5 or more might contribute to this goal.

Regards

Robert Smith

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