Here is the proof of two statements above. Just in case someone is interested.
It is easy to prove that the set of all prime numbers is infinite(more precisely - there is no greatest prime number). Here is how.
Let the set be finite and consist of {n1,n2,...nk}
Where nk is the largest prime number.
Then the number n1*n2*n3.....nk+1 does not belong to the above set. Also it is not divisible by either n1 or n2 or n3 ... or nk. So either it is prime or it has a prime factor larger than nk. In either case, we have found a prime number greater than nk.
This contradicts the assumption that nk was the largest prime number. So the assumption is false and there is no largest prime.
The second statement about all primes except for 2 and 3 being either 1 less or one more than 6. Here is why this is true.
Any positive natural number may be written in the form 6n+k. Where n is a natural number and k is one of {0,1,2,3,4,5}. A prime number cannot have k = 0 or k = 2 or k=4 because in all these cases, the number would be divisible by 2. It cannot have k = 3 because in this case it would be divisible by 3 as 6n+3 = 3*(2n+1). Thus it must be of the form 6n+1 or 6n+5=6(n+1)-1. So it must be one more or one less than a multiple of 6.