Prove Twin Primes Greater Than 3 Divisible by 12

Date: 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/