Prime TripletDate: 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/ |
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