Is p^2 + 2 always composite if p is a prime number greater than or equal to 5?

[blairebear]

Homework Statement

If p>=5 is prime, prove that p^2 +2 is composite.

Homework Equations

If we took p and divided it by 6 we would get remainder possibilities of 0, 1, 2,3,4,5

The Attempt at a Solution

p=6q p^2=36q^2 P^2=6(r) P^2+2=6r+2=2(3r+1) composite

p=6q+1 p^2=36q^2+12q+1 = 12(r)+1 p^2+2=12r+3 = 3(4r+1) composite

similarly 6q+2, 6q+4, 6q+5

But 6q+3 p^2= 36q^2+36q+9 = 9(4q^2)+4q+1) p^2=9r+2

?? How do I show this one is composite?

 

[blairebear]

Or can I just say p would not be prime since p would be divisible by 3? But then that also holds of 6q, 6q+2 and 6q+4. The only possible primes are 6q+1 and 6q+5 which I proved are composites. OK I think I am done.