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/