20060204, 12:21  #1 
Jun 2003
Oxford, UK
19·103 Posts 
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^nk. I think the limitation k<2^(n1) 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. http://hjem.get2net.dk/jka/math/simultprime.htm names the type in the tables. A common 4?/8/12/16/? name, e.g. "multiproth", would be nice for reference. Have you considered 4sets (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^nk 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 Last fiddled with by robert44444uk on 20060204 at 12:23 
20060204, 14:33  #2 
Nov 2003
2·1,811 Posts 
After several attempts (over half an hour) I was able to open his page. The current records are (using ith prime for our multiProths):
8primes: 206 digits (tuplet), Octoproth (n=346): 105 digits 12primes: 44 digits (bitwin), Dodecaproth (n=91): 28 digits 16primes: 26 digits (tuplet), Hexadecaproth (if we find one for n=76): 24 digits The hendicap of using the ith 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 17digit k. Then, we'll have 2 primes with 23 digits, 6 primes with 24 digits and 8 primes with 3941 digits which I think (obviously?) is better than 16 primes with 26 digits, but based on 8th prime with 24 digits our Hexadecaproth will be ranked below the 26digit tuplet. Another possible size measure I was thinking about is a "score" used by prof. Caldwell on Top5000 (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^(n1), since we begin the search from k=1, not from k=2^n1, and thus all our large multiProths satisfy the condition. Last fiddled with by Kosmaj on 20060204 at 14:35 
20060204, 14:34  #3 
Jun 2003
Oxford, UK
19·103 Posts 
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 
20060204, 15:56  #4  
"Robert Gerbicz"
Oct 2005
Hungary
5×13×23 Posts 
Quote:
Another note: Now I'm writing a program to search 22 primes in arithmetic progression. The world record is 23 primes. 

20060204, 18:45  #5 
Jun 2003
Oxford, UK
19·103 Posts 
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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