List all primes p below 10000 such that every permutation of p's digits is a prime (e.g. 2, 37, 199).
Any conjectures?

As digits are to be permuted, units digits cannot be 0, 2, 4, 5, 6 or 8 the exceptions of 2 and 5 as a single digit primes.
That leaves just 1,3,7 and 9 for use for multi-digit permuted primes.

These are all of the primes below 10000 using those digits (note the 2 and 5 exceptions):

1-digit:
2 3 5  7    
2-digit
11  13  17  19  31  37  71  73  79  97
3-digit
113  131  137  139  173  179  191  193  197  199
311  313  317  331  337  373  379  397  719  733
739  773  797  911  919  937  971  977  991  997
4-digit
1117  1171  1193  1319  1373  1399  1733  1777  1913  1931
1933  1973  1979  1993  1997  1999  3119  3137  3191  3313
3319  3331  3371  3373  3391  3719  3733  3739  3779  3793
3797  3911  3917  3919  3931  7177  7193  7331  7333  7393
7717  7793  7919  7933  7937  7993  9133  9137  9173  9199
9311  9319  9337  9371  9377  9391  9397  9719  9733  9739
9791  9931  9973

This table summarises the permutations by category for digit length (A,B,C & D are generalisations):
Len        1       2             3
Cat        A      AA  AB     ABC  AAB
Perm      1      1    2         6     3

Len         4
Cat      ABCD    AABC   AABB  AAAB
Perm      24        12         6        4

From the numeral list above only:
2 3 5 7
11  13/31  17/71  37/73 79/97
113/131/311   337/373/733  and 199/919/991
have all permutations for their respective category.

(The numeric list is the compilation of computer data lists of 2-, 3- and 4-digits with the single digit primes added.  My findings were manually derived).

Posted by brianjn on 2012-06-17 21:14:22