1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
  4. Jan 2, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook