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 CannOfPrimes 2013-09-17 18:29

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

 Batalov 2013-09-17 18:48

Only Generalized Fermat numbers (in base 10 in this case), can be prime:
10[SUP]N[/SUP]+1, where N=2[SUP]n[/SUP].

And even these are very rarely prime. It is conjectured that there's only a finite number of them.

You can see the start of the series [URL="http://factordb.com/index.php?query=10%5En%2B1&use=n&n=1&VP=on&VC=on&EV=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=20&format=1&sent=Show"]here[/URL] and [URL="http://www1.uni-hamburg.de/RRZ/W.Keller/GFN10.html"]here[/URL].

 gd_barnes 2013-09-18 11:44

[quote]
Can we be certain that all theorized PalPrimes of this nature are actually Prime.
[/quote]

To clarify further:

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=2^24-1.

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.

 gd_barnes 2013-09-18 21:40

Serge,

I have a new long-term 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) :smile:

Gary

 Batalov 2013-09-18 22:25

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 [I]after[/I] that. I am fine with only me knowing the true character of those numbers, though. ;-)

 gd_barnes 2013-09-18 23:39

[QUOTE=Batalov;353360]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 [I]after[/I] that. I am fine with only me knowing the true character of those numbers, though. ;-)[/QUOTE]

Oh wow, that is impressive. Let me see if I understand you correctly: You have tested both 10^(2^24)+1 and 10^(2^25)+1 and have proven that they are composite and you did this in 2008/2009? (!!) The latter one is over 33M digits and must have taken months, especially that long ago.

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.

 Batalov 2013-09-19 01:49

These are not hard problems. You put
[B][Worker #1]
PRP=1,10,33554432,1[/B]
in worktodo.txt and