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 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 $p^2 + 2$ to be a prime number is if $p^2$ is divisible by 3, since if it was not divisible by 3, it would be equivalent to 1 mod 3 and then $p^2 + 2$ would be divisible by 3, hence showing that it is not prime. But if $p^2$ is divisible by 3, then p = 3 since p is prime and we have $p^3 + 2= 3^3 + 2 = 29$, 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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