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Math Help - Prime Question

  1. #1
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    Prime Question

    For which positive integers n is n^4 + 4^n prime?

    I know 1 works. I know it can't be an even n because that would end up even, and thus not prime.

    I checked 3, 5, and 7 and they all don't work. I don't know how to prove this as working or not though for bigger odd numbers. I feel like 1 might be the only answer, but not sure how to prove that other numbers do/don't work.
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  2. #2
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    Quote Originally Posted by Janu42 View Post
    For which positive integers n is n^4 + 4^n prime?

    I know 1 works. I know it can't be an even n because that would end up even, and thus not prime.

    I checked 3, 5, and 7 and they all don't work. I don't know how to prove this as working or not though for bigger odd numbers. I feel like 1 might be the only answer, but not sure how to prove that other numbers do/don't work.
    You can also eliminate non-multiples of 5 when you consider mod 5. This leaves only n that are congruent to 5 mod 10.

    Not sure how to deal with those.. but the first 200 of them are composite.

    Maybe you can factor them. Let f(n) = n^4 + 4^n. There might be a pattern here: (note: f(25) is not given as prime factorization, rather I multiplied some primes to get two factors close together.)

    f(5) = 17 * 97
    f(15) = 29153 * 36833
    f(25) = 33350257 * 33759857
    f(35) = 34350564553 * 34368914633

    Difference between the two factors shows a pattern:

    80 = 2^4 * 5
    7680 = 2^9 * 3 * 5
    409600 = 2^14 * 5 * 5
    18350080 = 2^19 * 5 * 7
    Last edited by undefined; September 30th 2010 at 05:49 PM.
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  3. #3
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    Since we're dealing with n odd we can factor this way:

    <br />
n^4+4^n = (n^2+2^\frac{n+1}{2}n+2^n)(n^2-2^\frac{n+1}{2}n+2^n)<br />

    And for n>1 both factors are greater than one - the left factor is

    obvious. As for the second one - for the odds 3, 5 it can be checked

    by hand. For n\ge 7 we have

    n<2^\frac{n-1}{2} \Rightarrow 2^\frac{n+1}{2}n<2^n \Rightarrow 2^n-2^\frac{n+1}{2}n > 0 \Rightarrow n^2+2^n-2^\frac{n+1}{2}n > 1
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