How to make a pair of six-sided dice whose sum is always a prime number? In other words, how can I find two sets of six distinct integers $a_1, \dots, a_6 \in \Bbb Z$ and $b_1, \dots, b_6 \in \Bbb Z$ such that $a_i+b_j$ is prime for any $i, j \in \{1, \dots, 6\}$?
 A: All the answers here are much larger than necessary. I wrote a computer program to find a set of numbers such that the largest number used is as small as possible. An optimal solution is:
$$A = \{5, 13, 17, 47, 73, 83\}$$
$$B = \{0, 6, 24, 54, 66, 84\}$$
However this has duplicate possible rolls. If this is unacceptable and each combination of rolls must be distinct an optimal solution is:
$$A = \{0, 26, 56, 96, 140, 180\}$$
$$B = \{11, 17, 71, 83, 137, 173\}$$
A: A 12-term arithmetic progression of primes exists by Green-Tao (as stated in comments). Richard K. Guy, 1994 gives one as $23143 + 30030k$. So we can find two sets $A, B$:
$$A=\lbrace{23143, 53173, 83203, 113233, 143263, 173293\rbrace}$$
$$B=\lbrace{30030, 60060, 90090, 120120, 150150, 180180\rbrace}$$
A: We can have $$A=\{1,7,13,17,19,23\} \\
B=\{2670,24090,198810,560220,603900,663960\}$$
I just wrote a program to check multiples of $30$ for the second die that would work with the first using the base 2 Fermat test.  I then checked them by hand in Alpha, which disqualified $126210$ because $126217$ is a 2-pseudoprime.  This produces $36$ distinct prime numbers from the rolls.
