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A/b + b/a equals an integer, for any a,b nonzero positive integers, if and only if a

  1. Apr 10, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only if a = b

    2. Relevant equations

    Divisibility: d|e implies e = dj where j is an element of j for d and e element of the integers.

    3. The attempt at a solution

    1. Let a/b + b/a = k where k is an element of the integers

    2. a/b + b/a = (a^2 + b^2) / ab

    3. (a^2 + b^2) = kab

    4. Let (a^2 + b^2) = c where c is an integer

    5. c = kab

    6. Step five implies the three following things: a|c , b|c , ab|c

    7. If a|c and b|c, then ab does not divide c and therefore step 6 demonstrates a contradiction.

    8. Therefore a must equal b

    Is this a solid proof? If not, could you tell me where the weakness is?

    EDIT : If a and b are relatively prime. I cannot use my contradiction then! Any ideas ?

    Thank you for your time.
     
    Last edited: Apr 10, 2012
  2. jcsd
  3. Apr 10, 2012 #2
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    One thing that positive integer could be odd or even.
     
  4. Apr 10, 2012 #3
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Eh what if a and b are relatively prime. I cannot use my contradiction then! Any ideas ?
     
  5. Apr 10, 2012 #4
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    mtayab1994, what do you mean with that?
     
  6. Apr 10, 2012 #5
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    sorry my earlier comment was a typo i wanted to say if they were prime like you just stated
     
  7. Apr 10, 2012 #6
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Yeah, I thought about that after I typed it. Any idea on how to solve this?
     
  8. Apr 10, 2012 #7
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Well it seems that it really doesn't matter if they're prime, because take for example 5 ( a prime number). In this case we get c=a^2+b^2=50 so c/a=10 and c/b=10 and c/ab=2; and this will work for any number and that number c divided by ab is the same as a/b+b/a. Hopefully you can something in this.
     
  9. Apr 10, 2012 #8

    Office_Shredder

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    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    I don't understand the contradiction. If a and b divide c ab can definitely divide c as well. In fact I don't see anything about a=b until after you got a contradiction already.

    At step 3 just consider the equation mod a and b
     
  10. Apr 10, 2012 #9
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Yes that's what i tried showing him in the previous reply.
     
  11. Apr 10, 2012 #10

    Dick

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    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    I'm a little vague on what you think the contradiction is. I would assume a and b are relatively prime to begin with. (Otherwise you could just remove the common factors). Now suppose p is a prime divisor of a. What could you conclude from (a^2 + b^2) = kab?
     
  12. Apr 10, 2012 #11
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Relatively prime does not mean that they are prime. Relatively prime means that the gcd(a,b) = 1. They have no common factors. My contradiction fails if a and b have no common factors.

    Office Shredder, if a|c and b|c, then ab|c if gcd(a,b) = 1. Otherwise it does not work.
     
  13. Apr 10, 2012 #12
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Would that mean that p|(a^2 + b^2) ?
     
  14. Apr 10, 2012 #13
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Yes.
     
  15. Apr 10, 2012 #14
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Where does that take me?
     
  16. Apr 10, 2012 #15

    Dick

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    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    If there are multiple people responding it's a good idea to hit the Quote button before you reply to make it clear who you are responding to. But yes, assume gcd(a,b)=1 and try to derive a contradiction with that.
     
  17. Apr 10, 2012 #16
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    I'm not so sure.
     
  18. Apr 10, 2012 #17
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    So,
    1. a^2 + b^2 = kab

    2. Let gcd(a,b) = 1

    3. Let p be a prime divisor of a

    4. Then p|a^2 + b^2

    ...

    I'm thinking but it is not clicking.
     
  19. Apr 10, 2012 #18

    Dick

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    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Can't you show p must divide b as well?
     
  20. Apr 10, 2012 #19
    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    Would this work?

    a^2 + b^2 = pj where j is an element of the integers
    a = pw where w is an element of the integers

    Then:
    p^2w^2 + b^2 = pj

    we reorganize to:
    b^2 = p(j - pw^2)

    we let j - pw^2 = v where v is an element of the integers

    so b^2 = pv

    that implies p|b^2 and b times b will not create a factor p since p is prime therefore,

    p|b
     
  21. Apr 10, 2012 #20

    Dick

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    Re: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only i

    That's a little more complicated than it needs to be but if p|a then p|a^2 and p|kab, so p|b^2. And if p is prime, then sure, p|b. Doesn't that contradict gcd(a,b)=1?
     
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