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Math Help - Number Theory

  1. #1
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    Number Theory

    There do not exist positive integers x,y,z satisfying 2xz=y^2 & x+z=997
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  2. #2
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    Re: Number Theory

    Is there a...question?
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  3. #3
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    Re: Number Theory

    Proceed by using these two equations to form a quadratic equation. Since we have two equations and three variables we cannot have a unique solution. we will get infinitely many solutions.
    from first one we have z = (y^2)/(2x) plug in this in equation 2, we get
    x + (y^2)/(2x) = 997
    Form this quadratic and then try and reason out
    Thanks from swarna
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  4. #4
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    Re: Number Theory

    Quote Originally Posted by swarna View Post
    There do not exist positive integers x,y,z satisfying 2xz=y^2 & x+z=997
    Suppose x is even
    x=2t
    4\text{tz}=y^2
    t=u^2 , z=v^2 since t and z are relatively prime
    2u^2+v^2=997 which is impossible (Use congruence mod 8)
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