Primes from powers of 2 strings.

[ATH]

I tested the 2 strings without "shuffling":

2ⁿ..2⁰:
Only prime is n=2: 421 for n<636 (61106 digits)

2¹..2ⁿ2⁰:
Two primes for prime n=2: 241 and n=4: 248161. No others for n<656 (65200 digits)

I also tested how many of the n! numbers are prime for small n:

n=2 (3 digits): 2 primes of 2 numbers
n=4 (6 digits):11 primes of 24 numbers
n=6 (10 digits): 104 primes of 720 numbers
n=8 (16 digits): 4254 primes of 40320 numbers
n=10(23 digits): 260732 primes of 3628800 numbers.

[Flatlander]

Originally Posted by ATH

I tested the 2 strings without "shuffling":

2ⁿ..2⁰:
Only prime is n=2: 421 for n<636 (61106 digits)

Is it provable that it's the only one? (Maybe it's a pointless exercise anyway.)
All I could find was this: http://tech.groups.yahoo.com/group/p.../message/19065

Originally Posted by ATH

...
I also tested how many of the n! numbers are prime for small n:

n=2 (3 digits): 2 primes of 2 numbers
n=4 (6 digits):11 primes of 24 numbers
n=6 (10 digits): 104 primes of 720 numbers
n=8 (16 digits): 4254 primes of 40320 numbers
n=10(23 digits): 260732 primes of 3628800 numbers.

Much more than I would have guessed when I started this thread.

[axn]

Originally Posted by Batalov

I used perl to manipulate strings (pregenerated with BigInt), sans '1'. With an array of strings, just swap two random strings at a time, concat (with '1' at end) and pipe into pfgw. Just my 2c.

While you're at it, it is probably a good idea to use the lexically largest string as the first one, which will yield bigger primes for a given combination.

[Batalov]

Done that last night. There were just two small primes: 842161, 8644322161.

That and unpermuted concat as other people have done. Nothing new up to n~=750.
Which is as expected: a finite set, and a very short one, at that.
(421 is in both sets)

[retina]

Originally Posted by CRGreathouse

It's all in how you count it: is #88 2^0 through 2^88 or 2^0 through 2^87?

The OP defines n.

Originally Posted by Flatlander

Create primes by arranging the strings produced from progressive powers of 2 from 2^0 to 2^n.

[axn]

Originally Posted by Batalov

Done that last night. There were just two small primes: 842161, 8644322161.

No. I don't mean sort _all_ of them. Just pick the largest one of the set to be the prefix. Permute the rest as usual. That way, your prime would be one of the biggest for the given n. It is a cheap optimization to get a minor increase in prime size.

[CRGreathouse]

Originally Posted by retina

The OP defines n.

Yes, and I could have been using that n or n = |S|. It's not like I looked back at the OP before writing such a straightforward comment.

[Batalov]

Originally Posted by axn

No. I don't mean sort _all_ of them. Just pick the largest one of the set to be the prefix. Permute the rest as usual. That way, your prime would be one of the biggest for the given n. It is a cheap optimization to get a minor increase in prime size.

Yep, quite right, easily done.
And with that in mind, a prp20624 (n=368).
It had very few permutations away from a lexicographically reverse-sorted starting point.