Prove Twin Primes Greater Than 3 Divisible by 12Date: 10/08/2002 at 19:35:53 From: Crystal Subject: Prove twin primes>3 are divisible by 12 Prove that if p and q are twin primes, each greater than 3, then p+q is divisible by 12. I have looked in several books, but most of them use mod and we have not covered this topic yet. Date: 10/08/2002 at 19:47:20 From: Doctor Paul Subject: Re: Prove twin primes>3 are divisible by 12 First notice that all primes except two and three are either one more or five more than a multiple of six. A proof of this fact is in the Dr. Math archives: Primes Greater Than/Less Than Multiples of Six http://mathforum.org/library/drmath/view/56068.html An equivalent way of wording the above statement is that all primes are either one, five, seven, or eleven more than a multiple of twelve. If p is one more than a multiple of twelve, then p+2 is three more than a multiple of twelve, and in this situation p+2 will never be prime (since a number that is three more than a multiple of twelve will always be divisible by three). So if we want p and p+2 to both be prime, we cannot have p be one more than a multiple of twelve. A similar argument shows that if p is seven more than a multiple of twelve, then p+2 will never be prime. Thus the only way that p and p+2 can both be prime must occur when p is five more than a multiple of twelve or when p is eleven more than a multiple of twelve. If p is five more than a multiple of twelve, then p+2 is seven more than a multiple of twelve. In this case, p + p+2 is twelve more than a multiple of twelve, which is in fact a multiple of twelve. If p is eleven more than a multiple of twelve, then p+2 is one more than a multiple of twelve. In this case, p + p+2 is twelve more than a multiple of twelve, which is in fact a multiple of twelve. Thus whenever p and p+2 are prime, p + p+2 is a multiple of twelve. 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/ Date: 10/08/2002 at 19:48:26 From: Doctor Shawn Subject: Re: Prove twin primes>3 are divisible by 12 Crystal, Here's a hint to start -- prove that all twin primes greater than three must be of the form 6n + 1 and 6n - 1. I think the proof shouldn't be too hard after that! Hope that helps. If you're still stuck, write again. - Doctor Shawn, The Math Forum http://mathforum.org/dr.math/ |
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