When you multiply a red number, a green number and a blue number together, the resulting product is white . I believe the following are true. The numbers are here arranged in an anticlockwise spiral starting from 1 at the center. The above two, imply that any odd number not divisible by 3 is either the same colour it would be if prime or the complementary colour. These theorems are useful for proving much of what follows.
All square numbers are black, because the product of any two numbers of the same colour is black.
If neither a nor b is divisible by 3, the number is red . If either a or b is divisible by 3, but not both, the number is black . If both are divisible by 3, the number is divisible by 9, the quotient, has the same colour and is is a sum of two squares. A black or a red prime number is the sum of two squares. A black or a green prime number is the sum of two squares and their geometric mean.