# Math Help - Prime numbers

1. ## Prime numbers

Find all primes $p$ such that the number of $(p^6)+6$ is also a prime number.
How to begin to solve this problem?

2. ## Re: Prime numbers

The way I solved this problem was to use Wolfram Alpha: Plug in expressions of the form

Factor p^6 + 6

where p = 2, 3, 5, 7, 11, 13, 17, 19 and so on until you see a pattern. Then use what you know (perhaps Fermat's Little Theorem) to prove your guess. Others might be able to see the solution more quickly, but that's how I did it.

3. ## Re: Prime numbers

Hint: Consider $p^6 + 6$ modulo some other prime q. Without much computation, I was able to find a q such that $p^6 + 6 \equiv 0 (\mod q)$ for all p.