Find largest left truncatable prime in a given base

Obviously the right most digit must be prime, so in base 10 candidate are 2,3,5,7. Putting a digit in the range 1 to base-1 in front of each candidate must result in a prime. So 2 and 5 like the whale and the carnations in Hitchickers come into existance only to be extinguished before they had time to realize it. 13,23, ... 83,97 are candidates. Again Putting a digit in the range 1 to base-1 in front of each candidate must be a prime. Repeating until there are no larger candidates finds the largest left truncatable prime.

Find largest left truncatable prime in a given base is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

ref 1: https://oeis.org/A103443

related task 1: http://rosettacode.org/wiki/Truncatable_primes

Let's work base 3 by hand:

0 and 1 are not prime so the last digit must be 2. 12 = 5 which is prime, 22 = 8 which is not so 12 is the only candidate. 112 = 14 which is not prime, 212 = 23 which is, so 212 is the only candidate. 1212 = 40 which is not prime, 2212 = 76 which also is not prime. So there are no more candidates, therefore 23 is the largest left truncatable prime in base 3.

The task is to reconstruct as much, and possibly more, of the table in ref 1 as you are able.

Ruby

<lang ruby>

1. Compute the largest left truncatable prime
3. Nigel_Galloway
4. September 15th., 2012.

require 'prime' BASE = 3 MAX = 500 stems = Prime.each(BASE-1).to_a (1..MAX-1).each {|i|

 t = []
 b = BASE ** i
 stems.each {|z|
   (1..BASE-1).each {|n|
     c = n*b+z
     t.push(c) if c.prime?
 }}
 break if t.empty?
 stems = t

} puts "The largest left truncatable prime #{"less than #{BASE ** MAX} " if MAX < 500}in base #{BASE} is #{stems.max}" </lang> By changing BASE from 3 to 13 this produces:

The largest left truncatable prime in base 3 is 23

The largest left truncatable prime in base 4 is 4091

The largest left truncatable prime in base 5 is 7817

The largest left truncatable prime in base 6 is 4836525320399

The largest left truncatable prime in base 7 is 817337

The largest left truncatable prime in base 8 is 14005650767869

The largest left truncatable prime in base 9 is 1676456897

The largest left truncatable prime in base 11 is 2276005673

The largest left truncatable prime in base 13 is 812751503

The maximum left truncatable prime in bases 10 , 12, and 14 are very large. By changing MAX from 6 to 10 (note that 6 solves related task 1) the following are produced quickly in with BASE = 10:

The largest left truncatable prime less than 1000000 in base 10 is 998443
 
The largest left truncatable prime less than 10000000 in base 10 is 9986113

The largest left truncatable prime less than 100000000 in base 10 is 99979337

The largest left truncatable prime less than 1000000000 in base 10 is 999962683

The largest left truncatable prime less than 10000000000 in base 10 is 9987983617

The largest left truncatable prime less than 100000000000 in base 10 is 999783972
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