[QUOTE=science_man_88;235325]is there anyway other than vectors that i can test strings of x and y representing numbers and print the biggest one ?[/QUOTE]

I use binary() to generate a vector, then sum appropriately. There are other ways, of course, but essentially all the time is spent on other parts so it doesn't matter how long this part takes.

[QUOTE=3.14159;235335]Using 5 and 7;

75757575757575757575757575757575757575757575757575757575757575757575757575757 is prime. That might not have any prime substrings, beside 757.[/QUOTE]

Well, there's 75757575757575757.

I've had an example since yesterday that works (non-01 I mean), but I left it at home. Unfortunately I haven't yet been able to find the smallest one, so I won't be able to stick to my usual style here. I'm close to finding the smallest example with the two digits I happened to choose, though -- I have a list of all possible residues mod 10^20 for those digits.

Anyway; Is there any proof for the interesting property of odd numbers I may have posted earlier on?

If 2 and 5 do not divide n, then n^(10^n) = 1 mod (10^(n+1)).

Ex:

3^(10^2) = 515377520732011331036461129765621272702107522001 = 1 mod 10^(2+1).

7^(10^2) = 3234476509624757991344647769100216810857203198904625400933895331391691459636928060001 = 1 mod 10^(2+1).

Hint: 10[SUP]n[/SUP] | 4 * 10[SUP]n[/SUP].

[QUOTE=CRGreathouse;235362]Hint: 10[SUP]n[/SUP] | 4 * 10[SUP]n[/SUP].[/QUOTE]

This provides nothing useful.

[QUOTE=3.14159;235262]I was referring to primes that only use two digits.

Ex: 6666166666166661166116661, uses only 6 and 1. (This fails. Reason: 61, 661, 6661)

It's probable that every combination with a digit coprime to the other will always have a prime substring, except for primes that use 1 and 0 (As long as "11", or "101" does not appear, or for that matter, any "binary" prime.)

That is; That is, 2 and 1, 3 and 1, 4 and 1, 5 and 1, 6 and 1, 7 and 1, 8 and 1, 9 and 1, 3 and 2, 5 and 2, 7 and 2, 9 and 2, 4 and 3, 5 and 3, 7 and 3, 5 and 4, 7 and 4, 9 and 4, 6 and 5, 7 and 5, 8 and 5, 9 and 5, 7 and 6, 8 and 7, 9 and 7, 9 and 8, will all have a prime substring, given that it's over 15-20 digits. (Or will you be willing to provide a counterexample?)[/QUOTE]

5155555155551555551555551555551555551.

[QUOTE=3.14159;235368]This provides nothing useful.[/QUOTE]

Hint 2: You're raising the base to a much higher number than the number you're reducing by.

[QUOTE=Batalov;235280][pedantic]The 'covering set' for 4-and-9s is 449, 499, 9949, 94949. [/pedantic]

The first two primes do not exclude the third, but after the third the only suriving pattern is 949...49, but unfortunately it immediately dies out with 94949.[/QUOTE]
All of the two-digit combinations have very short 'covering sets' (Hint: the shortest sets are '2', '3', '5', '7', '11', and so on), and the only non trivial series is the 0-and-1-ers (with the covering set of '11' and '101' for the primes of form 10[SUP]a[/SUP]+10[SUP]b[/SUP]+10[SUP]c[/SUP]+1. Hint: 10[SUP]n[/SUP]+1 are Generalized Fermats and there are only two known primes of this form; because we are searching much lower than the current search limit for [URL="http://www1.uni-hamburg.de/RRZ/W.Keller/GFN10.html"]GFN10[/URL]'s, we are safe in this toy search and the exclusion check is very easy, as I've written earlier).

I am too lazy to provide all covering sets.

[QUOTE=Batalov;235379]All of the two-digit combinations have very short 'covering sets' (Hint: the shortest sets are '2', '3', '5', '7', '11', and so on)[/QUOTE]

1 digit primes are allowed, so more than just 01 primes are possible, e.g., 5155555155551555551555551555551555551.

Well, that contradicts the original definition - "no prime substrings" (of any length). That's too many forks on the road - for my taste. I am searching for large members in [URL]http://oeis.org/classic/A033274[/URL] with near-repdigit property.

Also, there's no such sequence "Primes that do not contain any other primes as a substring [I]longer than one digit*[/I]" (it would have contained [URL="http://oeis.org/classic/?q=1009%2C1021%2C1049%2C1051%2C1063&sort=0&fmt=0&language=english"]1009,1021,1049,1051,1063[/URL] )... but this of course can be solved, too, per se, and with near-repdigit property.

*or insert another description of an exemption

I found the smallest {1, 5} prime with over 20 digits: 115555515555555155551. Because the method I used to find this is written in two languages it's not convenient for me to look for the smallest {a, b} prime over 20 digits (at this point I'd have to run each search 'manually').

This one contains 11, though. If the substring condition is dropped, then it will be hard to beat 10[SUP]232599[/SUP]-3 and 5·10[SUP]211029[/SUP]-1