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Math Help - integers

  1. #1
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    integers

    Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.
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    Hello,
    Quote Originally Posted by mpryal View Post
    Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.
    Fact : For any integer m, m^2 \equiv 0 \text{ or } 1 (\bmod 4)

    Proof :
    A number m can only be even or odd.
    If it's even, it can be written m=2k, in which case, m^2=4k^2 \equiv 0 (\bmod 4)
    If it's odd, it can be written m=2k+1, in which case, m^2=4k^2+4k+1 \equiv 1 (\bmod 4)


    Now, is it possible that u=a^2+b^2 \equiv 3 (\bmod 4) when a^2 and b^2 can only \equiv 0 \text{ or } 1 (\bmod 4) ?
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