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{Number Theory} Smallest integer solution

  1. Aug 15, 2015 #1
    1. The problem statement, all variables and given/known data
    Let ##x,y,z## be positive integers such that ##\sqrt{x+2\sqrt{2015}}=\sqrt{y}+\sqrt{z}## find the smallest possible value of ##x##

    2. Relevant equations
    Not even sure what to ask i'm trying to learn number theory doing problems and look up information by doing the problems.



    3. The attempt at a solution

    The only thing I have done so far is get it set equal to ##x## so that I could make do some algebra and the answer would pop up.
    ##x=2\sqrt{yz}+y+z-2\sqrt{2015}##

    Thanks for any help.
     
  2. jcsd
  3. Aug 15, 2015 #2

    Dick

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    Should be kind of clear where to go from there. ##x,y,z## are integers. ##2\sqrt{2015}## is irrational. Something else irrational must cancel it. What must it be?
     
  4. Aug 15, 2015 #3
    But if it cancels it out wouldn't that make ##x=0## which I can't have?
     
  5. Aug 15, 2015 #4

    Dick

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    There are only two things with can be irrational in your expression. What the other one? Equate them.
     
  6. Aug 15, 2015 #5
    Thank you I'm little tired so I just didn't understand what you said the first time even though it was clear. I got the correct answer now with that hint which is ##96##
     
  7. Aug 16, 2015 #6

    Dick

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    You're welcome. But my argument why is pretty dodgy. If you want to be more rigorous, can you show that ##2015yz## must be a perfect square? What would that imply?
     
    Last edited: Aug 17, 2015
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