Ah, I must have missed the bit about the zeroes and ones. That should cut down the possibilities a lot.

I wrote my code in PARI/gp, and it's fairly dense and hard to understand so the runtime would be quicker. When I get back I'll have a see about updating the filtering of it and spreading it out a bit. I should be able to hugely cut down the run time with this extra information, especially with a bit of a re-write to take better account of the digit sum being 43.

I essentially just hacked in a little extra code to verify the digit sum onto existing code in a very sub-optimal way, but it still cut the run time from nearly 6 hours to 17.5 mins.

[QUOTE=Xyzzy;387728][CODE]×××××××××××××××× is not prime
×××××××××××××××_ is not prime
××××××××××××××__ is not prime
×××××××××××××___ is not prime
××××××××××××____ is not prime
×××××××××××_____ is not prime
××××××××××______ is not prime
×××××××××_______ is not prime
××××××××________ is not prime
×××××××_________ is not prime
××××××__________ is not prime
×××××___________ is not prime
××××____________ is not prime
×××_____________ is not prime
××______________ is not prime
×_______________ is not prime[/CODE][CODE]×××××××××××××××× is not prime
_××××××××××××××× is not prime
__×××××××××××××× is not prime
___××××××××××××× is not prime
____×××××××××××× is not prime
_____××××××××××× is not prime
______×××××××××× is not prime
_______××××××××× is not prime
________×××××××× is not prime
_________××××××× is not prime
__________×××××× is not prime
___________××××× is not prime
____________×××× is not prime
_____________××× is not prime
______________×× is not prime
_______________× is not prime[/CODE](Is there an easier way to say all of that without listing each example?)[/QUOTE]

I believe it could be summed up as the card number is the concatenation of two composite numbers.

Composite: 87654321
Composite: 12345678

Card Number: 8765432112345678

Truncate: 876543211_______

Prime: 876543211

[QUOTE=lavalamp;387886]Composite: 87654321
Composite: 12345678

Card Number: 8765432112345678

Truncate: 876543211_______

Prime: 876543211[/QUOTE]

okay so that one's out I was going to avoid adding in 15 different ways.

You only need 7 primality tests anyway:
EEEEEO
EEEEEOO
EEEEEOOO
EEEEEOOOEO
EEEEEOOOEOEEO
EEEEEOOOEOEEOO
EEEEEOOOEOEEOOO

And once you have the first 14 digits nailed down, you automatically know the last 2* due to the luhn check.

* Actually there are 2 possibilities for the last 2 digits when you know the first 14, but that's better than the 5*3=15 combinations it could be (since the last digit can't be 0 or 2).

[QUOTE=science_man_88;387881]I believe it could be summed up as the card number is the concatenation of two composite numbers.[/QUOTE]

No, it could not. 7333333773333337 is the concatenation of two composite numbers (73333337 twice), but it contains primes in the portions described.

I could restate it more concisely than Xyzzy did, but probably not in a way that is easier or clearer than his statement. Here goes...

The full number is C, a 16-digit composite number.
For n=0 to 15 (inclusive), [TEX]C \bmod 10^n[/TEX] (this covers the statement about the rightmost digits) and [TEX]{C - (C \bmod 10^n)}\over{10^n}[/TEX] (and this is the leftmost digits) are composite.

[QUOTE=Mini-Geek;387844]Xyzzy: let us know if your number is really on that list![/QUOTE]It is not in your list. Please see our next post for more info.

[QUOTE=Xyzzy;387751]There are no situations where a 0 or a 1 are alone in any of those listings.[/QUOTE]We are not certain if this "hint" is the problem, but to be safe, we are going to retract it because we are not sure it says what we want it to say.

We have checked several times to make certain that our previous statements are absolutely correct.

We will tell our card number to Ernst who will verify our hints, just to make sure.

Once that has been verified, we may offer additional hints.

We are so stupid.

:gah:

[CODE]_______________× [U][B]is[/B][/U] prime[/CODE]

We have emailed Ernst the request to verify our hints, just to make sure we have not made any other blunders.

We apologize for the error!

:max:

[QUOTE=Xyzzy;388025]We are so stupid.

:gah:

[CODE]_______________× [U][B]is[/B][/U] prime[/CODE]

We have emailed Ernst the request to verify our hints, just to make sure we have not made any other blunders.

We apologize for the error!

:max:[/QUOTE]

Assuming this was the only blunder on either of our parts, I've created a [URL="https://drive.google.com/file/d/0B72y7-pdB73beEN0WWRCTmhZUGM/view?usp=sharing"]new list with 691,157 numbers[/URL]. Just a hair fewer numbers than before. I'm guessing the sum of 43 is what accounts for nearly as many ending with 2 as 4, 6, and 8 combined.

So it ends in a 2?