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Math Help - Infinite solutions proof

  1. #1
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    Infinite solutions proof

    Hello all,


    I'm practicing proofs and I'm stuck. Here it is:


    Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2 + y^2 = z^2. Evidently I'm supposed to start by setting x, y, and z like this:


    x = m^2 - n^2
    y = 2mn
    z = m^2 + n^2


    So then we have:


    (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2
    m^4 + n ^4 - 2(mn)^2 + 4(mn)^2 = m ^4 + n^4 +2(mn)^2
    m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2 . . . Both sides of the equation are equal
    2(mn)^2 = 2(mn)^2 . . . Subtract m^4 and n^4 from both sides.
    2 = 2 . . . Divide both sides by (mn)^2

    Since 2 always equals 2 there are infinite solutions to the equation m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2. Because x^2 + y^2 = z^2 equals m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2, x^2 + y^2 = z^2 also has infinite solutions.

    Is this a valid proof or am I missing something?
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  2. #2
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    Re: Infinite solutions proof

    A much easier solution:

    For example, (x,y,z) = (3,4,5) works. We can multiply x,y,z by the same integer constant yielding (x,y,z) = (6,8,10), (9,12,15), etc. Hence there are infinitely many solutions.
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  3. #3
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    Re: Infinite solutions proof

    I'm supposed to use this method. The only part I'm unsure of is where I derive 2=2. Did I do this properly?
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  4. #4
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    Re: Infinite solutions proof

    Ah okay. Your solution is correct, but too long. You can just stop when you know both sides of the equation are equal, i.e. m^4 + n^4 + 2(mn)^2 = m^4 + n^4 + 2(mn)^2. This equality implies that the above holds regardless of your choice of m and n, as long as m > n (so that x is positive).
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  5. #5
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    Re: Infinite solutions proof

    So, I guess I could say that there are infinite solutions because both sides of the equation are identical. That is, as long as m > n.
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