Tuesday, February 22, 2011

Find all primes p such that...

(1) 2^p + p^2 is also prime.

(2) 3^p + p^3 is also prime.

6 comments:

  1. Any others? If not, could you prove that there are no other solutions?

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  2. Actually, I have no idea. Tell me why there are no any other solutions.

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  3. p=2 gives us a composite
    p=3 gives us a prime number
    So, now, we examine p > 3.
    assume with p > 3 we get a prime number.
    Then p is odd.
    Write p = 2k+1.
    Then 2^p = 2^(2k+1) = 2*4^k = 2 (mod 3)
    since p is not equal to 3,
    p^2 = 1 (mod 3)
    hence
    2^p + p^2 = 2 + 1 (mod 3)
    = 3 (mod 3)
    = 0 (mod 3)
    There it cannot be a prime number.

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  4. With (2) use the same line of reasoning

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