# Math Help - Prime numbers

1. ## Prime numbers

Prove that if both p and p^2 + 8 are prime numbers, then p^3 + 10 is also prime

thanks.

2. If $p=3$, then $p^2+8=17$ which is again prime, and surely enough $p^3+10=37$ is prime.

Now if $p\neq 3$ - and p prime-, then $p^2\equiv{1}(\bmod.3)$ -since p must be coprime to 3- thus $p^2+8\equiv{0}(\bmod.3)$ so $p^2+8$ cannot be prime now.

Then, since $p$ and $p^2+8$ are primes simultaneously only for $p=3$, and in this case the assertion is true, we are done.