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Larsen problem

  1. Jan 2, 2008 #1
    [SOLVED] larsen problem

    1. The problem statement, all variables and given/known data
    Determine all integral solutions of [itex]a^2+b^2+c^2=a^2 b^2[/tex]. (Hint: Analyze modulo 4.)


    2. Relevant equations



    3. The attempt at a solution
    a^2,b^2,c^2 are congruent to 0 or 1 mod 4 implies that a^2,b^2,c^2 are all congruent to 0 mod 4. This implies that a,b,c are even.

    [tex]a=2a_1, b=2b_1, c=2c_1[/tex]

    Then we have [itex]a_1^2+b_1^2+c_1^2 = 4a_1^2 b_1^2[/itex]. Now it is very clear that a_1^2,b_1^2,c_1^2 are all congruent to 0 mod 4.

    Let [itex]a_1=2a_2,b_1=2b_2,c_1=2c_2[/itex].

    If we keep doing this, we get 3 decreasing sequences of positive integers that never reach zero, which is impossible.

    Therefore there are no solutions.

    Is that right?
     
  2. jcsd
  3. Jan 2, 2008 #2
    anyone?
     
  4. Jan 2, 2008 #3

    Dick

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    Is there some element of this proof that you aren't confident of? Because I don't see anything to worry about.
     
  5. Jan 2, 2008 #4
    No. I'm just not confident in my proofs in general and the word "all" in the problem statements made me think there would be at least one.
     
  6. Jan 2, 2008 #5

    Dick

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    Well, there is a=0, b=0 and c=0. But you knew that, right?
     
  7. Jan 2, 2008 #6
    Of course :uhh:

    The reason my proof does not apply to that case is because then, for example, a,a_1,a_2,... is constant sequence, nondecreasing sequence of 0s. However, if any of a,b,c are nonzero then everything in my proof applies.
     
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