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Math Help - Prime numbers

  1. #1
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    Prime numbers

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

    thanks.
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  2. #2
    Super Member PaulRS's Avatar
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    Oct 2007
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    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.
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