Interesting properties of prime numbers

All primes > 3 are of the form (6n + 1) or (6n - 1) where n is a natural number. Therefore the square of a prime will be of the form (36n^2 + 12n + 1) or (36n^2 - 12n + 1). The "interesting" properties pretty much follow from that.

The author must be a blast at parties:

"Ask your friends to pick any prime number bigger than 5, but they must not tell you what it is. Square it."

Sorry, couldn't resist. Lovably geeky.

I wonder, are these tricks proven for all primes?

Do we even know all primes?

No, we're still missing three of them. Has anyone checked behind the couch?

Euclid's theorem demonstrates that there is an infinite number of primes, so it's impossible to know all of them.

We don't, but cats do...

Yes they can be ( http://www.murderousmaths.co.uk/games/primcal.htm, http://primes.utm.edu/notes/faq/six.html ) but the unfortunate thing is that while the ((p^2)-1 % 24 == 0) test is a necessary requirement for primality testing, it is not sufficient. Lots of non-primes like 25, 49, 155 etc. also have 0 remainders.

Such properties would be proven given only the assumption of primality. People don't actually deal with prime numbers themselves.