# Find answer of $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$

If I had a very shallow question, then I am sorry. $x,y,z\in\mathbb{N}^{+}$ and$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$$find $x,y,z$.

I try with AM-GM, just get$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq\frac{3}{2}$$

This means that the equation must have a real solution, but can not be sure there is an integer solution.

Let: $x=ay=abz$, then the equation becomes:$$\frac{ab}{a+b}+\frac{b}{ab+1}+\frac{1}{a^2+ab}=4$$

Which makes the problem become non-homogeneous, and seems to become more difficult. I have no more ideas. Could anyone help me? Thanks a lot.

• If $y+z=a$ etc. $$2(4+1+1+1)=(a+b+c)\left(\dfrac1a+\dfrac1b+\dfrac1c\right)$$ – lab bhattacharjee Mar 18 '17 at 17:14
• Maybe a track using $\frac{x}{y+z} = \frac{x+y+z}{y+z}-1$, I get the equation $(x+y+z)^2=7(x+y)(y+z)(x+z)$. Then I suppose that $7$ divides $x+y+z$. – M. Boyet Mar 18 '17 at 17:19
• I wrote a small brute force program and find no solutions with $x,y,z \le 1000$ – Ross Millikan Mar 18 '17 at 17:40
• See the Online Encyclopedia of Integer Sequences oeis.org/A283564, where one comment says the first solution has 81 digits. – Michael Mar 19 '17 at 5:10
• This should help: mathoverflow.net/questions/227713/… – Mike Miller Mar 25 '17 at 12:43

There are indeed positive integer solutions. The smallest solution with $x, y, z$ in $\mathbb{N}_+$ is
$\small x = 154476802108746166441951315019919837485664325669565431700026634898253202035277999\\ \small y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579\\ \small z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036$
Well, why not? Multiplying through by denominators leads to cubic surface $$x^3 + y^3 + z^3 - 3 \left(x y^2 + x^2 y + y z^2 + y^2 z + z x^2 + z^2 x \right) - 5xyz = 0$$ which has integer points $$(1,1,-1),$$ $$(11,4,-1),$$ $$(11,9,-5)$$ and perhaps no others except for permuting and changing all signs. Or multiplying by any common integer factor.