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Math Help - Prime number

  1. #1
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    Prime number

    If 1+2^n+4^n is prime then prove that n is 3^k.Both variables are natural numbers.
    Help would be appreciated.
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  2. #2
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    Re: Prime number

    What have you tried so far? My gut is telling me to use the division algorithm n = 3^k q + r. Then since the order of 2 in the multiplicative group of \mathbb{Z} / 3^{k+1}\mathbb{Z} is known to be (3-1)\cdot 3^{k+1-1} = 2\cdot 3^k, you can further use the division algorithm on 4^{3^kq+r} = 4^r\left(4^{3^k}\right)^q = 4^r\left(1+3^{k+1} x\right)^q for some x \in \mathbb{Z}. Maybe an induction argument will arise that will yield that if n \neq 3^k then 3 divides 1+2^n + 4^n.
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