[adgjlsfhk1]

you want the nth bit to be set because otherwise there is a small but noticable chance that you generate a surprisingly weak prime.

[opticfluorine]

Out of curiosity, if it is known that the nth bit is set, don't I also have the same risk but in (n-1) bits? Genuinely curious here.

Edit: Ah, nevermind, I see now why I don't have that issue. It's because I can't easily iterate the primes in that domain even though I can iterate the reduced number of bits. Thanks!