Sieved?
 

I'm wondering if Cunningham numbers on the 3- and 3+ tables have been completely factored for small factors up to a specific size. In particular, I'm wondering if something like the following is true:

All numbers of the form 3^n-1 for 1000<n<100,000 must have a factor larger than 10^40.

[QUOTE=Zeta-Flux;85641]I'm wondering if Cunningham numbers on the 3- and 3+ tables have been completely factored for small factors up to a specific size. In particular, I'm wondering if something like the following is true:

All numbers of the form 3^n-1 for 1000<n<100,000 must have a factor larger than 10^40.[/QUOTE]

Unfortunately, numbers of this size could be tested to this sort of limit only through ECM, which can only place a probability that factors of that size have been discovered. I suspect you need some sort of rigorous limit, which is beyond present-day computing possibilities.

I suspect that many numbers in this range may have not been tested beyond 25 digits, judging from my experience with the 2- and 2+ tables.

Actually 25 digits would still be interesting (if it is true that they've all been tested up to this level).

[QUOTE=Zeta-Flux;85641]I'm wondering if Cunningham numbers on the 3- and 3+ tables have been completely factored for small factors up to a specific size. In particular, I'm wondering if something like the following is true:

All numbers of the form 3^n-1 for 1000<n<100,000 must have a factor larger than 10^40.[/QUOTE]Around a year or so ago some of the Cunningham old-timers estimated how many factors under P40 were left in the tables. We all knew, of course, that we wouldn't know the answer for sure until all the numbers had been completely factored.

I certainly didn't win the competition. I forget the precise details but my guess was something like 2 remaining and 3 were duly found.

The consensus is that it is now very unlikely that there are any prime factors with fewer than 40 digits still to be discovered. None have been found this year. The smallest reported is P43 and there are 30 in my table for 2006 which arel under P50. There may be one or two mpre P4x to be added when I bring my table up to date.


Paul

[QUOTE=Zeta-Flux;85641]I'm wondering if Cunningham numbers on the 3- and 3+ tables have been completely factored for small factors up to a specific size. In particular, I'm wondering if something like the following is true:

All numbers of the form 3^n-1 for 1000<n<100,000 must have a factor larger than 10^40.[/QUOTE]Er ... those aren't Cunningham numbers. The 3+ and 3- Cunningham tables only go as far as n=600 (1200 for the 3LM numbers).

I do not know of any tabulations of factorizations of the numbers in question. It would be easy enough to create such tables if they don't exist. If any are in existence, I doubt they would be complete much beyond p20 or so.



Paul

xilman,

If they are not techinically Cunningham numbers (yet), they are in the same spirit.

But, I would even be happy to know they have been checked up to 20 digits. The reason I ask is that, with the search for Mersenne primes, we do a lot of sieving for small factors. I was wondering if the same is done for the 3 tables (up to large exponents).