I would think that with the great computational skills evident
on this forum that the following derivation would be considered excessive:
Let P
p be the probability that a random prime p is a number n.
By the lemma, P
p = P
p-1 = ... .
Hence multiplying the P
p gives
(P
p)
n --> 1 or 0
depending on whether there exist any primes.
Additional Lemma: There are primes!
Proof: Start counting at 1 and continue until a number is reached
whose only factors are (well, you know). This process terminates at p=2.
Hence there are primes!
Corollary: The desired probability is 1 (if there really are primes).
