20130216, 18:15  #1 
May 2004
New York City
2×3^{2}×5×47 Posts 
Primes Concatenated
Consider the oeis sequence
2, 23, 235, 2357, 235711, 23571113, etc. of integers formed by concatenating the first n primes. Now derive the Mersenne subsequence 2^21, 2^231, 2^2351, 2^23571, 2^2357111, etc. It's unlikely any of these but the first will turn up prime (in that final reckoning). But if M2357 hasn't been factored, it ought to be. It has 710 digits. 
20130216, 18:30  #2  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×5×19×47 Posts 
Quote:
That number has been quadruple checked. The GIMPS database has 3 checks, one by nonPrime95 software. This gent proved it composite http://neoview.kicksass.net/mersenne/ Dario's ECM app returns it as composite in a flash. 

20130217, 00:17  #3 
May 2004
New York City
2×3^{2}×5×47 Posts 
I was just referring to the fact that 2357 is a pretty gem
and so 2^23571 deserves a special cutting (factoring). The next two numbers in the series, 2^2357111 and 2^23571113 are not yet completely factored either but leave apparently rather big composites. I didn't realize how much work had already been done here, so I wasn't reacting to a lack of interest. 
20130217, 01:24  #4  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
21736_{8} Posts 
Quote:


20130217, 09:43  #5  
Jun 2003
7·167 Posts 
Quote:
Turns out that an awful lot of effort has already been expended on your number. It's highly unlikely to have a factor below 55 digits, and will probably be beyond even the most wellresourced team to factor for the next several decades. But that shouldn't stop you from trying. Go run some more curves on it. And thank George Woltman for making this such an easy thing for you to do. 

20130217, 14:22  #6  
May 2004
New York City
1000010000110_{2} Posts 
Quote:
expended on 2357, and I was just suggesting it as an interesting fourdigit exponent that might be addressed, not (as I acknowledged in my second post) necessarily worth the extra effort ATP. Hey I'm as grateful as anyone for the work being done here. 

20150102, 20:09  #7 
May 2004
New York City
2×3^{2}×5×47 Posts 
What makes a prime interesting? Generally, it must have some special form. The Mersenne primes are
repunits in binary. The primes of the form in this thread's OP are formed by concatenating sequences of primes then forming the corresponding Mersenne number and factoring or proving primality. Etc. Etc. Forming new primes by concatenating other primes and factoring just seemed like a nice idea to try. So if no factors of 2^23571 are known, I'd like to call its smallest such factor (for no particular reason) a "cat" prime. 
20150119, 18:42  #8 
May 2004
New York City
2·3^{2}·5·47 Posts 
Perhaps we can relate the home primes to other concatenation of primes sequences
and get some notsolelyarithmetic results out of all this work and computation. 
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