p, p+8, p+22 Not Prime

Date: 10/16/2001 at 07:37:17
From: Scott S
Subject: Proof On Prime Numbers

Could you help me with this question?:

Prove that there is no positive integer p such that each of the 
numbers:

p
p+8
p+22

is prime.

Any help would be much appreciated.
Cheers,
Scott

Date: 10/16/2001 at 08:38:49
From: Doctor Paul
Subject: Re: Proof On Prime Numbers

If p is not prime, we're done. So assume p is prime. Then p is either 
one more than a multiple of three, or p is two more than a multiple of 
three.

In other notation:

p = 1 mod 3 or p = 2 mod 3.

   If p = 1 mod 3 then p+8 = 0 mod 3 and is therefore divisible by 
   three and is hence not prime.

   If p = 2 mod 3 then p+22 = 0 mod 3 and is therefore divisible by 
   three and is hence not prime.

Write back if you want to talk about this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/