1089 isn't prime either (1089 is 9*121). I think this approach might not work in general because of the two relevant numbers for the first die needing to share a factor. The first prime has to be of the form A+B, where A= 2022/(prime divisor of 2022), and B = number that is a multiple of 2022/(product of 2 its primes) for the above method to work. But no such prime exists since A must be divisible by two of (2,3, 337), and B must also be divisible by two of (2,3, and 337), so they must share at least one non-one factor, and can't be prime.