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Interesting Question

  1. Feb 2, 2006 #1

    JasonRox

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    This isn't homework, but I'll post it anyways because I'd like to know.

    It's from Herstein's Abstract Algebra.

    Show that if u = 4n + 3, where [itex]n\inN[/itex], then you can not write u in the from u = a^2 + b^2, where [itex]a,b\inN[/itex].

    I feel silly for asking this, but I'm curious to know.

    The one thing I do, which is obvious is that if a is odd, then b is even because u is odd. But I don't think you need this fact to solve it.

    Any directions?

    Please do not post solutions!!!
     
  2. jcsd
  3. Feb 3, 2006 #2

    VietDao29

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    Okay, I'll give you a hint:
    So one of a, and b must be odd, and the other is an even number, right?
    So let a = 2k, b = 2x + 1 (k, x are all integers).
    Now what's a2 + b2? If you divide a2 + b2 by 4, what's the remainder?
    You can take it from here, right? :)
     
  4. Feb 3, 2006 #3

    JasonRox

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    That's exactly what I did!

    I knew something wasn't right when I was looking at it.

    I'll give it another shot thanks.
     
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