Primes Concatenated

[davar55]

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

2^2-1, 2^23-1, 2^235-1, 2^2357-1, 2^235711-1, 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.

[Uncwilly]

Originally Posted by davar55

But if M2357 hasn't been factored, it ought
to be. It has 710 digits.

Why?
That number has been quadruple checked. The GIMPS database has 3 checks, one by non-Prime95 software. This gent proved it composite http://neoview.kicks-ass.net/mersenne/

Dario's ECM app returns it as composite in a flash.

[davar55]

I was just referring to the fact that 2357 is a pretty gem
and so 2^2357-1 deserves a special cutting (factoring).
The next two numbers in the series, 2^235711-1 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.

[Batalov]

Originally Posted by davar55

The next two numbers in the series, 2^235711-1 and 2^23571113 are not yet completely factored either but leave
apparently rather big composites...

http://oeis.org/A069151 are the only interesting ones.

[Mr. P-1]

Originally Posted by davar55

if M2357 hasn't been factored, it ought
to be. It has 710 digits.

Go factor it, then. That's what I did when I decided, for reasons no worse than yours, that M1369 "ought" to be factored. I didn't come here and suggest that other people "ought" to work on my pet project.

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 well-resourced 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.

[davar55]

Originally Posted by Mr. P-1

Go factor it, then. That's what I did when I decided, for reasons no worse than yours, that M1369 "ought" to be factored. I didn't come here and suggest that other people "ought" to work on my pet project.

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 well-resourced 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.

I was posting in ignorance of how much work has already been
expended on 2357, and I was just suggesting it as an interesting
four-digit 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.

[davar55]

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 2^2357-1 are known, I'd like to call its smallest such factor (for no particular reason)
a "cat" prime.

[davar55]

Perhaps we can relate the home primes to other concatenation of primes sequences
and get some not-solely-arithmetic results out of all this work and computation.