Primes: p+1 a Multiple of 6?
Date: 10/06/2002 at 00:18:10 From: Ryan Subject: Proving a statement (involving primes) Prove that if p and p+2 are both prime, then p+1 is divisible by 6.
Date: 10/06/2002 at 12:54:58 From: Doctor Paul Subject: Re: Proving a statement (involving primes) This statement isn't true. If p = 3, then p+2 = 5 is also prime, but p+1 = 4 is not a multiple of six. This is the only case for which the statement fails to be true. If we assume p > 3, then we can proceed as follows: First notice that all primes 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 If p is one more than a multiple of six, then p+2 is three more than a multiple of six and will never be prime (since a number that is three more than a multiple of six 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 six. Thus the only way that p and p+2 can both be prime must occur when p is five more than a muliple of six. If p is five more than a multiple of six, then p+1 is six more than a multiple of six. This is just another way of saying that p+1 is a multiple of six. 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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