Quote:
Originally Posted by Thomas11
Note that your formula (k = k + covering set * y) yields only 1/103680th of the k's of the cycle (one per cycle of length "covering set", but each cycle by itself consists of "n" k's). There is another recursion to get the other ones:
k = (2*k + (covering set)/2) (mod covering set)
This also means that your k is actually not the smallest k of the cycle.
In your case k0=27877812802424205899986999806089.

I know. This was just the first example. I stopped after I generated the first few. There are other ways of generating these numbers... you can generate a few TB of these numbers.
I don't think the size of the k will make any difference as all the k's generated will be low weight. Also k=64 bit would not be faster than these larger k's.
Since these numbers take too long to LLR I turned the problem around... and took k numbers less than 2^18. Then I split the k into base 2^5760 sequences.
k1=k*2^1
k2=k*2^2
... and so on.
I tested their weight using the above covering set (for 2^57601) & sieving to 1200
Only a few k remained with less than 20 candidates up to 1 million (corresponding to nash weight of 1/4). I have tested some of these to 2 Million with no prime.
There are more left if any one is interested. These can be sieved using srsieve. Any thoughts of a group project?