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Math Help - I need to prove that a term of a certain form cannot be a perfect square.

  1. #1
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    I need to prove that a term of a certain form cannot be a perfect square.

    This is part of a much larger problem.
    I need to prove that:
    \forall n\in\mathbb{Z}, n^2 \neq (4k+3),\forall k\in\mathbb{Z}

    Please help me in any way you can. Thanks.
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  2. #2
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    Hint - Take n = 4m + 1 or 4m+3 ? What do you see? You don't need to consider even 'n'
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  3. #3
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    Hello, miatchguy!

    Prove that: . \forall n\in\mathbb{Z},\: n^2 \neq (4k+3),\:\forall k\in\mathbb{Z}

    \,n must be either even or odd: . \begin{Bmatrix} n \:=\: 2m \\ \text{or} \\ n \:=\: 2m+1 \end{Bmatrix}


    If \,n is even: . n^2 \;=\;(2m)^2 \;=\;4m^2
    . . \,n^2 is a multiple of 4.


    If \,n is odd: . n^2 \;=\;(2m+1)^2 \;=\;4m^2 + 4m + 1 \;=\;4(m^2+m)+1
    . . \,n^2 is one more than a multiple of 4.


    Therefore, \,n^2 is never three more than a multiple of 4.
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