October 7th, 2013, 09:30 AM  #1 
Newbie Joined: Sep 2012 Posts: 25 Thanks: 0  prime number
p and p^2 + 2 are prime numbers. prove that p^3 + 2 is prime number too. 
October 7th, 2013, 10:59 AM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: prime number
The only way for to be a prime number is if is divisible by 3, since if it was not divisible by 3, it would be equivalent to 1 mod 3 and then would be divisible by 3, hence showing that it is not prime. But if is divisible by 3, then p = 3 since p is prime and we have , which is prime.

October 7th, 2013, 11:01 AM  #3 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Re: prime number
Try looking at this mod 3.

October 7th, 2013, 11:29 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,881 Thanks: 1088 Math Focus: Elementary mathematics and beyond  Re: prime number
pē + 2 = (p  1)(p + 1) + 3. All primes greater than 3 are of the form 6n  1 or 6n + 1, hence pē + 2 is prime only when p = 3.

October 7th, 2013, 12:29 PM  #5 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Re: prime number
I was trying for a subtle hint, though it wasn't that subtle...


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