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Math Help - even numbers

  1. #1
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    even numbers

    An evenly even number is a number in the form of 2^m , where m is a positive integer.
    Prove that it is impossible the sum of two evenly even numbers to be a perfect square.
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  2. #2
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    but 2^3+2^3=16 is a perfect square
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    As is 2^5+2^2=6^2. Not a very good conjecture!

    By the way the real name for an "evenly even number" is power of two.
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  4. #4
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    In fact, I believe 2^m+2^n is a perfect square if and only if either:
    i) both m and n are odd and m=n, or
    ii)either m or n is even (say m is even) and n=m+3.

    So, the example I gave was part (i) and the example Bruno J. gave was part (ii).

    Proof.
    The "if" part is pretty straight forward.
    So assume 2^m+2^n is a perfect square. And assume part i) doesn't hold. So we will show part ii) holds. Suppose, m \leq n.
    Then 2^m+2^n=2^m(1+2^(n-m)). This implies m is even and 1+2^(n-m) is a perfect square. Write 1+2^(n-m)=k^2. Then 2^(n-m)=(k-1)(k+1). This implies k-1 is a power of 2. Write k-1=2^p. Then 2^(n-m)=2^(2p)+2^(p+1). The right hand side of the last equality can be a power of 2 only if p=1. This implies 2^(n-m)=8 or n=m+3.
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