Prime Triplet

Date: 12/07/2001 at 16:44:38
From: Lynn
Subject: Prime triplets

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

My guess is that there is only one occurrence of such a prime triplet.  
I looked up to almost 2000 in a table of primes to see if there were 
any other occurrences, but there weren't.  If there is just the one 
case, how do I prove it?

Thanks,
Lynn

Date: 12/07/2001 at 16:54:53
From: Doctor Paul
Subject: Re: Prime triplets

3, 5, and 7 is the only such prime triplet. The proof is easy.  

Suppose x, x+2, and x+4 are prime and x > 3.  Well, x is not a 
multiple of three because if it were, then x would not be prime. 
So x is either one more than a multiple of three or two more than 
a multiple of three.

In the first case (x is one more than a multiple of three), x+2 will 
be a multiple of three and hence won't be prime (contrary to our 
assumption).

In the second case, x+4 will be a multiple of three - another 
contradiction.

Thus we have a contradiction in all situations, which means that the 
assumption must be invalid.

Thus 3, 5, 7 is the only such prime triplet.

I hope this helps.  Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/