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Homework Help: Integer solution

  1. Sep 7, 2010 #1
    1. The problem statement, all variables and given/known data

    let [tex]x,y \in N[/tex]. Solve the equation [tex]\frac{1}{x}+\frac{1}{y}=1[/tex]

    2. Relevant equations

    n/a

    3. The attempt at a solution

    so i can see x=y=2 is the solution, hmm, isn't there any other solution.

    and, this is one of three of my assignment question for 15% continuous assessment, does it seem too easy? should i prove that they are no other solution other than x=y=2?? help
     
    Last edited: Sep 7, 2010
  2. jcsd
  3. Sep 7, 2010 #2
    You shouldn't prove that there is no other solution because there are other solutions. How about x = 3 and y = (3/2)? or x = 6 and y = (5/6) ??? This is an implicit function describing a curve (in this case a straight line) of some sort, which means there are infinite points (values of x and y) that satisfy the equation. Try making it an explicit function by solving for y (or x if you want) and then see what you get.
     
  4. Sep 7, 2010 #3
    owh i forgot to wrote it, it's on the title "integer solution" hmm lemme edit
     
  5. Sep 7, 2010 #4
    Hint: If x > 2, then [itex]\frac{1}{x}[/itex] is ... .

    Petek
     
  6. Sep 7, 2010 #5
    If [tex]x>2[/tex] then [tex]\frac{1}{x}+\frac{1}{y}<\frac{1}{2}+\frac{1}{y}[/tex] and if [tex]y \geq 2[/tex], then [tex]\frac{1}{x}+\frac{1}{y}<\frac{1}{2}+\frac{1}{2}=1[/tex]

    similarly if [tex]y>2[/tex] and [tex]x \geq 2[/tex], so either case is not a solution. should i do like that?
     
  7. Sep 7, 2010 #6
    Yes, that's the right idea.

    Petek
     
  8. Sep 7, 2010 #7
    Petek illustrates an important method, but this diophantine equation can be solved explicitly. Simply clear denominators and rearrange to get xy - x - y = 0. Now try to add a constant to both sides so that you can factor the left hand side as a product of two linear factors, similar to completing the square.
     
  9. Sep 8, 2010 #8
    cool, i get

    [tex](x-1)(y-1)=1[/tex] implies [tex](x-1)=1 \ and\ (y-1)=1[/tex]

    but what if

    [tex]\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1[/tex], i can't solve it explicitly can't i?

    so i know[tex]x=y=z=3[/tex] is a solution

    and i need to verify, [tex]x \in N-(3)\ ,y \in N-(3)\ ,z \in N-(3)[/tex] is not a solution

    so for [tex]x=1,2[/tex] easy to verify

    if [tex]x>3 \ \ \ \ [/tex] [tex]\frac{1}{x}+\frac{1}{y}+\frac{1}{z}<\frac{1}{3}+\frac{1}{y}+\frac{1}{z}[/tex] so when y=1,z=2 and y=2,z=1 and y=1,z>3 and y=2,z>3 and y>3,z=1 and y>3,z=2, is not a solution(i have to verify all these cases?)

    and then repeat this tedious cases for y>3 and z>3

    is that the way?
     
  10. Sep 8, 2010 #9

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Actually, 1 factors in two different ways over the integers:
    1 = 1*1
    1 = -1 * -1
     
  11. Sep 8, 2010 #10
    owh yes, thank ;P

    so (x-1)=-1 =>x=0 similarly y=0, which is not a solution.

    but how about this


    is this argument correct? T_T
     
  12. Sep 8, 2010 #11

    hunt_mat

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    Homework Helper

    If
    [tex]
    \frac{1}{x}+\frac{1}{y}=1
    [/tex]
    Then:
    [tex]
    y=\frac{x}{x-1}
    [/tex]
    As y is an integer, we know that the above fraction is an interger, so x-1 must perfectly divide x, when can this happen?
     
  13. Sep 8, 2010 #12
    hmm since gcd(x-1,x)=1 so x/(x-1) to be integer means x-1 must equal to 1 or -1, right?

    hey, if i solve explicitly like this or like

    that's the only solution right? and other integer is not a solution?? and i don't need to verify others is not a solution right?? right right??? is it right?
     
  14. Sep 8, 2010 #13

    hunt_mat

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    Homework Helper

    Correct, but you can get rid of one of the options with a bit of thought. I was mainly thinking that if x is an integer then the greatest divisor must be x/2 if x is even or less if x is odd.
     
  15. Sep 8, 2010 #14
    wait what you mean by
    ?????


    and thanks for this clue
    i'm interesting to prove that statement ;P
     
    Last edited: Sep 8, 2010
  16. Sep 8, 2010 #15

    hunt_mat

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    Homework Helper

    I always loathed number theory. Look at the what x-1=1 and x-1=-1 and see what it means with the original equation.

    As with the rule, it was just an observation. I have no idea how to prove it.
     
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