Prime numbers

• November 4th 2012, 04:22 PM
makramer
Prime numbers
Quote:

Find all primes $p$ such that the number of $(p^6)+6$ is also a prime number.
How to begin to solve this problem? http://www.mymathforum.com/images/smilies/icon_wink.gif
• November 4th 2012, 05:02 PM
Petek
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.
• November 4th 2012, 06:36 PM
richard1234
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.