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October 7th, 2013, 08:30 AM   #1
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prime number

p and p^2 + 2 are prime numbers.
prove that p^3 + 2 is prime number too.
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October 7th, 2013, 09:59 AM   #2
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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.
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October 7th, 2013, 10:01 AM   #3
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Re: prime number

Try looking at this mod 3.
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October 7th, 2013, 10:29 AM   #4
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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.
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October 7th, 2013, 11:29 AM   #5
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Re: prime number

I was trying for a subtle hint, though it wasn't that subtle...
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