EDIT: To hurry along the conversation a bit - it's entirely possible I've been misled into thinking things are more simple than they really are. Wouldn't be the first time...
The question is why certain lines have lots of primes and not others? why is a given polynomial so rich in primes, while other similar ones are not?
That's what some of these conjectures are trying to prove.
Huh? If you draw the Ulam spiral on a checkerboard, all odd numbers end up on squares of one color and all even numbers of squares of the other color, but that is neither necessary nor sufficient to get those diagonal streaks in the picture.