Binary Palindromic Primes

CannOfPrimes · 2013-09-17, 18:29

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^N+1, where N=2ⁿ.

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 here and here.

gd_barnes · 2013-09-18, 11:44

Quote:

Can we be certain that all theorized PalPrimes of this nature are actually Prime.

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)

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

gd_barnes · 2013-09-18, 23:39

Quote:

Originally Posted by Batalov

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. ;-)

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
[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 2^383838383-1, so this is tiny compared to that.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Binary Multitasking	a1call	Lounge	8	2016-12-03 21:20
Happy Palindromic day!	ET_	Lounge	4	2012-02-21 18:20
Would you use a 'fat binary' of GMP-ECM?	jasonp	GMP-ECM	8	2012-02-12 22:25
2-d binary representation	only_human	Miscellaneous Math	9	2009-02-23 00:11
Palindromic Primes	S80780	Math	5	2003-04-13 23:49

2013-09-17, 18:29	#1
CannOfPrimes Sep 2013 1 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

2013-09-17, 18:48	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 10011101001011₂ Posts	Only Generalized Fermat numbers (in base 10 in this case), can be prime: 10^N+1, where N=2ⁿ. 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 here and here.

2013-09-18, 21:40	#4
gd_barnes "Gary" May 2007 Overland Park, KS 11854₁₀ Posts	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) Gary

2013-09-18, 22:25	#5
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×7×479 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. ;-)

2013-09-19, 01:49	#7
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 10059₁₀ 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 2^383838383-1, so this is tiny compared to that.