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Math Help - Elementary # Theory

  1. #1
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    Elementary # Theory

    Hey, guys, this might be a stupid question, but

    How do you prove that

    4K+3 can not be a perfect square, if K is any integer.

    Thanks a lot .
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  2. #2
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    All integers are of four different forms: 4x, 4x + 1, 4x + 2, and 4x + 3 for some integer x. Since squares of even numbers are even, 4x + 3 must be the square of an odd number, if it would be a square. All odd numbers are either of the form 4x + 1 or 4x + 3. Now show that (4x + 1)^2 and (4x + 3)^2 are both of the form 4x + 1, and thus 4x + 3 cannot be the square of any odd integer, and hence cannot be the square of any integer.
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  3. #3
    Moo
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    Hello,

    More simply, a number is either 2n or either 2n+1.

    (2n)^2=4n^2=4n' \neq 4k+3

    (2n+1)^2=4n^2+4n+1=4(n^2+n)+1=4n'+1 \neq 4k+3
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