[esruqntc] fun with the beginning of the primes

{0, 1, 2} are special.  find or create (interesting) theorems that are true precisely for each of the 8 possible subsets of those numbers unioned with the odd primes starting from 3.  then, claim each such theorem defines the primes.

inspired by someone uneducated attempting to troll mathematicians by asking whether 1 is a prime.  in reality, when the relevant set of numbers at beginning of the primes matters (say, for a theorem), mathematicians will uncontroversially state that assumption as part the theorem.  most often, things that are true about primes are interesting because they are true about the larger primes.  {0, 1, 2} are boring.

"prime", when classifying the numbers {0, 1, 2}, is a social construct, not ground truth.  it is a convenience.  the most commonly convenient classification is to exclude 0 and 1 and include 2, but if in some application it is convenient to use a different subset, just state that subset and move on.