Prove It

For my number theory class, we were assigned two problems on the first day. The problem that was most interesting was about prime numbers. A prime number is a number that is divisible by 1 AND itself. 2, 3, 5, 7, 11, 13 ... are prime numbers. The problem is this ...

The consecutive odd numbers 3, 5 and 7 are all primes. Are there infinitely many such "prime triplets?" That is, are there infinitely many prime numbers, p, such that p+2 and p+4 are also primes?

The answer is NO! But how do you prove it??? Dema to the rescue. Here is his proof ...

For all p excluding the number 3.

(Fig 1.) <-----p---p+1---p+2---p+3---p+4---->

1. Assume P is prime.
2. Assume there exists infinitely number of prime triplets of the form S = {p, p+2, p+4}.
3. Statement 2 implies that p+2 and p+4 are always prime for any prime number p..
4. If p is not divisible by 3, then p+1 or p+2 must be divisible by 3.

• Condition 1 - p is prime and p+1 is not divisible by 3 ... implies that p+2 is divisible by 3. Contradiction of statement 3.

• Condition 2 - p is prime and p+1 is divisible by 3 ... implies that p+4 is divisable by 3. Contradiction of statement 3.
  

Therefore, each condition cannot exists to statify assumption number 2 and there cannot be an infinitely many "prime triplets." ... So you're saying I have a chance!!!