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Math Help - How do you prove that there is no perfect square of the form 4k+3?

  1. #1
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    How do you prove that there is no perfect square of the form 4k+3?

    Im stuck on this homework problem that im not sure if i am doing right and it is due tommorow:
    Prove that there is no perfect square of the form 4k+3?

    Case 0:
    k is even
    k=2n for some n in Z
    4(2n)+3=8n+3 which is odd so there is no perfect square if k is even

    Case 1:
    k is odd
    k=2n+1 for some n in Z
    4(2n+1)+3 = 8n+7 which is not of the form 4k+3 there for there is no perfect square
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  2. #2
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    Re: How do you prove that there is no perfect square of the form 4k+3?

    I think you're a little confused.

    Squares of even numbers are even so 4k+3, which is odd of course, is not the square of an even number.

    Squares of odd numbers are of the form (2m+1)^2 = 4m^2+4m+1 = 4(m^2+m) + 1 which is of the form 4k+1.
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  3. #3
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    Re: How do you prove that there is no perfect square of the form 4k+3?

    Hahah i feel like such a derp!

    THANK YOU xxxx SPACIBAAAAAAAA !
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  4. #4
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    Re: How do you prove that there is no perfect square of the form 4k+3?

    Note: this solution's similar to a tutor's solution, but uses mods instead. Note that n must be odd. We have two cases:

    n \equiv 1 (\mod 4). Then n^2 \equiv 1 (\mod 4)

    n \equiv 3 (\mod 4). Then n^2 \equiv 1 (\mod 4).

    In either of these cases n^2 \not \equiv 3 (\mod 4), that is, it cannot be written in the form 4k+3, k integer. Hence we're done.
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