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Homework Help: A =/= a^2 + d^2 (abstr alg)

  1. Feb 3, 2013 #1
    1. The problem statement, all variables and given/known data
    All work is done in the set of integers, Z
    a is congruent to 3 modulo 4, Want to prove a=/= c^2 + d^2 for any c or d

    2. Relevant equations

    3. The attempt at a solution
    My work:
    a congruent to 3 modulo 4 => a -3 = 4k
    or a = 4k +3
    Then want to prove that: 4k+3 =/= c^2 + d^2 *** for any c or d

    I've tested out some experimental values for 4k+3 and indeed none of them have any c or d that match this equation but i'm not sure how to progress from *** onwards, It is probably a very simple matter but I can't wrap my head around this.
  2. jcsd
  3. Feb 3, 2013 #2


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    Science Advisor
    Homework Helper

    You can express ANY integer c as 4k, 4k+1, 4k+2 or 4k+3. What are the possible expressions for c^2? In other words, what are the possible values of c^2 mod 4?
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