Find all prime numbers $p$ such that $p^{2}+2007 p-1$ is prime as well.

Hint. Except for 3, prime numbers are not divisible by 3 . Hence if $p$ is not equal to 3 then either $p=3 k+1$ or $p=3 k-1$ for some integer $k$. If you wish you may use lists of prime numbers from the internet (e.g. www.imomath.com/primes
If $p=3$, then $p^{2}+2007 p-1=6029$ which is a prime. For $p \neq 3$, we know that $p=3 k \pm 1$ hence $p^{2}+2007 p-1=$ $9 k^{2} \pm 6 k+1+2007 p-1=9 k^{2} \pm 6 k+2007 p$ which is divisible by 3 and can't be prime. Thus the only such prime number is $p=3$.