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Math Help - Prime numbers

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
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    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.
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