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Thread: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square number.

  1. #1
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    X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square number.

    Okay, so I'm stuck on this for an hour now, so I'd really appreciate some help, please:

    X, Y, Z are coprime integers. We also know that 1/x + 1/y = 1/z. Prove that (X+Y) is a square number.

    -----
    This should be entry level university math, but I'm stuck with it. I hope I posted to the right topic. Thanks for the help.
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  2. #2
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    Re: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square numbe

    Are x, y, z pairwise coprime or coprime together?

    In mathematics, the case of variable names matters. For example, X and x are different variables.
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    Re: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square numbe

    I apologize. Please consider x=X and so on. My mistake.

    They are coprime together: The problem says that their greatest common divisor is 1.

    Thank you.
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    Re: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square numbe

    OK, here is a sketch. Suppose that p, q, r are prime, i, j₁, j₂ and k are positive integers, and $\displaystyle x=p^iq^{j_1}$, $\displaystyle y=q^{j_2}r^k$. In the general case, the variables here should be vectors. Without loss of generality, let $\displaystyle j_1\le j_2$. Then

    $\displaystyle z=\frac{xy}{x+y} =\frac{p^iq^{j_1+j_2}r^k}{q^{j_1}(p^i+q^{j_2-j_1}r^k)} =\frac{p^iq^{j_2}r^k}{p^i+q^{j_2-j_1}r^k}$.

    Let $\displaystyle s=p^i+q^{j_2-j_1}r^k}$. Then the only primes that could divide s are p, q and r. If p | s, then $\displaystyle p\mid q^{j_2-j_1}r^k$, a contradiction. Similarly, r | s is false. Thus $\displaystyle s=q^j$ for some j. Moreover, $\displaystyle j=j_2$; otherwise q ∈ GCD(x, y, z). Finally, if $\displaystyle j_2>j_1$, then q | s implies $\displaystyle q\mid p^i$, a contradiction. Therefore, $\displaystyle j_1=j_2=j$, $\displaystyle s=q^j$ and $\displaystyle x+y=q^{2j}$.

    See if you can shorten this and if it can be extended to the general case.
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  5. #5
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    Re: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square numbe

    Quote Originally Posted by emakarov View Post
    OK, here is a sketch. Suppose that p, q, r are prime, i, j₁, j₂ and k are positive integers, and $\displaystyle x=p^iq^{j_1}$, $\displaystyle y=q^{j_2}r^k$. In the general case, the variables here should be vectors. Without loss of generality, let $\displaystyle j_1\le j_2$. Then

    $\displaystyle z=\frac{xy}{x+y} =\frac{p^iq^{j_1+j_2}r^k}{q^{j_1}(p^i+q^{j_2-j_1}r^k)} =\frac{p^iq^{j_2}r^k}{p^i+q^{j_2-j_1}r^k}$.

    Let $\displaystyle s=p^i+q^{j_2-j_1}r^k}$. Then the only primes that could divide s are p, q and r. If p | s, then $\displaystyle p\mid q^{j_2-j_1}r^k$, a contradiction. Similarly, r | s is false. Thus $\displaystyle s=q^j$ for some j. Moreover, $\displaystyle j=j_2$; otherwise q ∈ GCD(x, y, z). Finally, if $\displaystyle j_2>j_1$, then q | s implies $\displaystyle q\mid p^i$, a contradiction. Therefore, $\displaystyle j_1=j_2=j$, $\displaystyle s=q^j$ and $\displaystyle x+y=q^{2j}$.

    See if you can shorten this and if it can be extended to the general case.
    To extend your ideas to the general case, may I suggest that we write x = da and y = db where d = gcd(x,y).
    From this we get dab=(a+b)z and using your method we show that d = a + b.
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  6. #6
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    Re: X, Y, Z are coprime integers. 1/x + 1/y = 1/z. Prove that (X+Y) is a square numbe

    Generally this equation quotient more general equation.

    equation: $\displaystyle XY+XZ+YZ=N$

    Solutions in integers can be written by expanding the number of factorization: $N=ab$

    And vospolzovavschis solutions of Pell's equation: $\displaystyle p^2-(4k^2+1)s^2=1$

    $\displaystyle k$ - what some integer number given by us.

    Solutions can be written:

    $\displaystyle X=ap^2+2(ak+b+a)ps+(2(a-2b)k+2b+a)s^2$

    $\displaystyle Y=2(ak-b)ps+2(2ak^2+(a+2b)k+b)s^2$

    $\displaystyle Z=bp^2-2(2b+a)kps+(4bk^2-2ak-b)s^2$

    And more:

    $\displaystyle X=-2bp^2+2(k(4b+a)+b)ps-2((4b+2a)k^2+(2b-a)k)s^2$

    $\displaystyle Y=-(2b+a)p^2+2(k(4b+a)-b-a)ps-(8bk^2-(4b+2a)k+a)s^2$

    $\displaystyle Z=bp^2-2(2b+a)kps+(4bk^2-2ak-b)s^2$

    Perhaps these formulas for someone too complicated. Then equation: $\displaystyle XY+XZ+YZ=N$

    If we ask what ever number: $\displaystyle p$ That the following sum can always be factored: $\displaystyle p^2+N=ks$

    Solutions can be written.

    $\displaystyle X=p$

    $\displaystyle Y=s-p$

    $\displaystyle Z=k-p$
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