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Math Help - help me solve this problem

  1. #1
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    help me solve this problem

    prove that if n is a perfect square, then n+2 is not a perfect square.

    I approached this way:
    hypothesis: n+2=k^2
    conclusion: n=k;

    But i tried both direct and indirect proof, but neither works. Or my approach is not right? pls help
    Last edited by zpwnchen; September 13th 2009 at 06:25 PM.
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  2. #2
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    Hello, zpwnchen!

    Prove that if n is a perfect square, then n+2 is not a perfect square.

    n is a perfect square: . n \:=\:a^2\,\text{ for some positive integer }n.

    Suppose n+2 is a perfect square: . n+2 \:=\:b^2\,\text{ for some positive integer }b.

    We have: . \begin{array}{c}n \:=\:a^2 \\ n \:=\:b^2-2 \end{array}

    Then: . a^2 \:=\:b^2-2 \quad\Rightarrow\quad b^2 - a^2 \:=\:2 \quad\Rightarrow\quad (b-a)(b+a) \:=\:1\!\cdot\!2


    We have a system of equations: . \begin{array}{ccc}b-a &=& 1 \\ b+a &=& 2\end{array}

    . . which has the solution: . a \:=\:\tfrac{1}{2},\:b \:=\:\tfrac{3}{2}


    But a\text{ and }b are integers . . . We have reached a contradiction.

    Therefore. n + 2 is not a perfect square.

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  3. #3
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    Thank you so much!

    Can we say that n=a^2 where a is 1/2. therefore, n is not a perfect. we reached a contradiction?

    Can we use direct or indirect proof for it?
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