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Squared fractions.

  1. Aug 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that [tex]1/72[/tex]
    cannot be written as the sum of the reciprocals
    of the squares of two different positive integers.


    2. Relevant equations



    3. The attempt at a solution

    Available solutions
    1/8²-1/24²
    1/9²+1/648
    Therefore Proven.
     
  2. jcsd
  3. Aug 14, 2009 #2
    [tex]\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{72}[/tex]

    [tex]\frac{b^2+a^2}{a^2b^2}=\frac{1}{72}[/tex]

    [tex]72(b^2+a^2)=a^2b^2[/tex]

    Solve the equation and write here the solution.
     
  4. Aug 14, 2009 #3
    How do i solve that equation?
     
  5. Aug 14, 2009 #4
    Move the terms from the left to the right side of the equation:

    [tex]a^2b^2-72b^2-72a^2=0[/tex]

    Now factor out b2 or a2 and tell me what you got.
     
  6. Aug 14, 2009 #5
    [tex]a^2=(72b^2)/(b^2-72)[/tex]
     
  7. Aug 14, 2009 #6
    Ok. Now what "a" is equal to? What can you conclude from the final solution?
     
  8. Aug 14, 2009 #7
    Thanks for your help. But the actual problem is to derive the available solution from the question given.
     
  9. Aug 14, 2009 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, that is what he is trying to show you how to do!

    However, the problem, as you stated it was

    "Show that 1/72 cannot be written as the sum of the reciprocals
    of the squares of two different positive integers." (my emphasis)

    You can't do that because, as you showed, it is not true.
     
  10. Aug 14, 2009 #9
    I got it. Im sorry.

    Now for part 2,
    How can I write [tex]1/72[/tex] with reciprocals of the squares of
    three different positve integers.
     
    Last edited: Aug 14, 2009
  11. Aug 14, 2009 #10

    Mark44

    Staff: Mentor

    Start by writing an equation that expresses this relationship.
     
  12. Aug 14, 2009 #11
    okay. [tex]1/a^2+1/b^2+c^3=1/72[/tex]
    By studying the relationship of their factor,
    The equation can be translated into :
    [tex]1/x^2+1/(b^2)(x^2)+1/(c^2)(x^2)=1/72[/tex]
    Moreover,

    [tex]b^2+c^2+1=x[/tex]
     
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