Which integers can be written in form $\frac {(x+y+z)^2}{xyz}$,where $ x,y,z $ are positive integers?

Asked May 10 '20 at 08:02

Active May 10 '20 at 08:02

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Question:Which integers can be written in form $\frac {(x+y+z)^2}{xyz}$,where $x,y,z$ are positive integers?

I suppose $ f(x,y,z)=\frac {(x+y+z)^2}{xyz}=n$

$nxyz=(x+y+z)^2$

$nxyz=x^2+y^2+z^2+2xy+2yz+2zx$

$nxyz=(x+y)^2+2(yz+zx)+z^2$

After that i got stuck, I have no idea how to proceed further.

asked May 10 '20 at 08:02

Paras

1

See Toronto Mathematics Competition. – Dietrich Burde May 10 '20 at 08:09
1

The answer to this post is also helpful. – Dietrich Burde May 10 '20 at 08:11
$\dfrac{(x+y+z)^2}{xyz}=n \implies \Bigl(2 (x + y + z) - n y z\Bigr)^2 + \Bigl(2 (y + z)\Bigr)^2 = \Bigl(n y z - 2 (y + z)\Bigr)^2$ – Dmitry Ezhov May 10 '20 at 12:18
look here please: https://artofproblemsolving.com/community/c6h1832412p12273114 – Pavel Kozlov May 10 '20 at 12:58
I gave a very careful answer at http://math.stackexchange.com/questions/1930438/equation-with-vieta-jumping-xyz2-nxyz/1930796#1930796 – Will Jagy May 10 '20 at 18:38

0 Answers0