Prime Chains

Problem #149

Tags: puzzle arithmetic

Who solved this?

Idea for this puzzle was proposed by Gaurav Kamath aka Moriarty - thanks a lot!
Note that it is really more about efficient data structure and algorithm rather than about math...

Look at the integer 997739719 - it has a funny property - any substring of 3 digits i.e.:

997739719
---------
997
 977
  773
   739
    397
     971
      719

any of these 997, 977, ..., 719 is a prime!

Such numbers built of chains of primes could be long enough, and they could be built using primes of more than 3 digits.
Of course it is important that prime substrings should be unique - since otherwise we can generate endless integers like 997399739... and so on.

For example let us search for integers built of 5-digit primes. Note that resulting chain could start with leading zeroes, like this:

0000233331793979193911399793199139313339119333773

since 00002 is a valid prime number too.

You will be given hashes of several such prime-chain-integers - and your task is to print these integers themselves. Since we are interested in long enough integers, please ignore any of them with length less than 48 digits (leading zeroes included).

Hash Calculation

We'll use the following simple algorithm to calculate the hash of a long NUMBER:

HASH = 0
for D in digits(NUMBER):
    HASH = (HASH * 13 + D) % 100000007
return HASH

where D are decimal digits picked from the NUMBER in order "left-to-right". For example the hash of the value 000233...333773 shown above is 34475206.


Input and Output

Input data contain the amount of numbers to search for.
Next lines contain a single hash value each.
Answer should contain long prime-chain-number (of the length 48 digits or more) corresponding to each hash.

Example:

input data:
3
34475206
90962736
39728928

0000233331793979193911399793199139313339119333773
    749031379939791939113997931991393133317939371999719
        461031379939791939113997931991393133317939371999719

Values here are placed into separate lines since they are just too long to fit a single line! However you should simply separate them with spaces.

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