+ We can narrow down the prime selecting process based on the divisble-by-three

+ principle (that is, that if the sum of the digits of a number is divisible by

+ three then the number is divisible by three).

+ That helps because any number that is divisible by three is not prime (that is,

+ any number but three itself).

+ The following are charts over the first few n replaced digits. The chart marks

+ an X where the sum will be a 3. The last column shows the number of possible

+ primes for this permutation. Using the charts I will show that you can only

+ ever have an 8 prime family when n is a multiple of three.

+ Y axis is sum of other digits mod 10.

+ Chart for 1 * replaced digits.

+ =================================

+ 0 1 2 3 4 5 6 7 8 9 Max

+ 5 ... pattern continues

+ Chart for 2 * replaced digits.

+ =================================

+ 0x2 1x2 2x2 3x2 4x2 5x2 6x2 7x2 8x2 9x2

+ 0 2 4 6 8 1 3 5 7 9 Max

+ 5 ... pattern continues

+ Chart for 3 * replaced digits.

+ =================================

+ 0x3 1x3 2x3 3x3 4x3 5x3 6x3 7x3 8x3 9x3

+ 0 3 6 9 3 6 9 3 6 9 Max

+ 0 X X X X X X X X X X | 0

+ 3 X X X X X X X X X X | 0

+ 5 ... pattern continues

+ Chart for 4 * replaced digits.

+ =================================

+ 0x4 1x4 2x4 3x4 4x4 5x4 6x4 7x4 8x4 9x4

+ 0 4 8 3 7 2 6 1 5 9 Max

+ 5 ... pattern continues