20130917, 18:29  #1 
Sep 2013
1_{10} Posts 
Binary Palindromic Primes
Can we be certain that all theorized PalPrimes of this nature are actually Prime.
10000.............000001 (uneven number of binary digits) 101 10001 1000001 
20130917, 18:48  #2 
"Serge"
Mar 2008
Phi(3,3^1118781+1)/3
10001101100101_{2} Posts 

20130918, 11:44  #3  
May 2007
Kansas; USA
10156_{10} Posts 
Quote:
At this time, we can be certain that only 2 of them are prime. The only known primes in this series are for n=1 and 2, i.e. 10^1+1 and 10^2+1 or simply 11 and 101. All n's have been tested up to n=2^241. In a synopsis, this means that all 10^n+1 where n<2^24 have effectively been tested for primality, which only required that 24 tests be run, i.e. n=2^0, 2^1, 2^2, 2^3, etc. up to 2^23. It is unlikely that there are any more primes in this series but there is no way to prove that with current technology and knowledge. Last fiddled with by gd_barnes on 20130918 at 21:32 

20130918, 21:40  #4 
May 2007
Kansas; USA
2^{2}×2,539 Posts 
Serge,
I have a new longterm project for you. I see that you were the one who proved that 10^(2^23)+1 is composite way back in 2008. How about you test to see if 10^(2^24)+1 is composite? (16777217 digits) Gary 
20130918, 22:25  #5 
"Serge"
Mar 2008
Phi(3,3^1118781+1)/3
13×17×41 Posts 
I actually have,  as well as all composite cofactors in the same range (for all five bases up to n<=25 iirc). I've done that back in 2008, 2009. (Note that you have to use base other than 3 for GFN'(3,n).)
I have reported them, too. But I guess, W.Keller didn't know me too well back then and wrote back that he will not post them just yet, and will wait for someone else to repeat that. Of course that was before I found three Fermat factors for him in a row. I think he might have trusted my results better if I submitted PRP test results after that. I am fine with only me knowing the true character of those numbers, though. ;) 
20130918, 23:39  #6  
May 2007
Kansas; USA
2^{2}×2,539 Posts 
Quote:
Assuming so, I can kind of see why he wants a doublecheck but I'm surprised that he doesn't change their "character" to composite (with perhaps a question mark besides it like GIMPS does until all doublechecks below a prime are complete) and show somewhere that a doublecheck is needed. 

20130919, 01:49  #7 
"Serge"
Mar 2008
Phi(3,3^1118781+1)/3
13×17×41 Posts 
These are not hard problems. You put
[Worker #1] PRP=1,10,33554432,1 in worktodo.txt and ThreadsPerTest=6 (or 8) in local.txt, and you will be done in no time on a computer with ECC memory. As you know, people already ran LL tests on 2^3838383831, so this is tiny compared to that. 
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