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Math Help - Prime number of a form

  1. #1
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    Prime number of a form

    Find all primes of the form 2^(2^n)+5, where n is a nonnegative integer

    Thanks very much!!
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  2. #2
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    Quote Originally Posted by suedenation View Post
    Find all primes of the form 2^(2^n)+5, where n is a nonnegative integer

    Thanks very much!!
    Only n=0
    Because if n>0 then you have
    none, all these numbers are divisible by three.
    Proof
    ~~~
    We know that,
    2 = -1 (mod 3 )
    Then,
    2^(2^n) = (-1)^(2^n) (mod 3)
    But,
    (-1)^(2^n)=1 because exponent is even.
    Thus,
    2^(2^n) = 1
    Add 5 to both sides,
    2^(2^n)+5 = 1+5=6=0 (mod 3)
    Q.E.D.
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