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Math Help - Primes

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    Primes

    Show that if m^4+4^n is prime, then m is odd and n is even, except when m=n=1.
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    Quote Originally Posted by eyke View Post
    Show that if m^4+4^n is prime, then m is odd and n is even, except when m=n=1.
    well, if m is even, then obviously 4 \mid m^4 + 4^n and so m^4 + 4^n cannot be prime. if n=2k+1, \ k \geq 1, then putting a=2^k we have 4^n=4^{2k+1}=4a^4.

    now m^4+4^n=m^4+4a^4=(m^2+2a^2+2ma)(m^2+2a^2-2ma) and so m^4+4^n cannot be prime because m^2+2a^2-2ma=(m-a)^2+a^2 \geq a^2 > 1.
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