A serious challenge:
Can someone find 3 positive whole numbers that solve this equation?
$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
The numbers must be whole!
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$\begingroup$ Do you have the answer? $\endgroup$ – CY Aries Jun 14 '17 at 17:22
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2$\begingroup$ what have you tried so far? $\endgroup$ – Dando18 Jun 14 '17 at 17:22
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2$\begingroup$ @soccerStack If you already know the answer, then this isn't the website to post a puzzle. You should post it on the puzzling SE website. $\endgroup$ – Χpẘ Jun 14 '17 at 18:20
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2$\begingroup$ Possible duplicate of Find answer of $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$ $\endgroup$ – Dietrich Burde Jun 14 '17 at 18:54
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1$\begingroup$ @soccerStack There is one given in the MO question which is linked at the duplicate, in the article "An Unusual Cubic Representation Problem" by Andrew Bremner. It has " truly enormous size". $\endgroup$ – Dietrich Burde Jun 14 '17 at 19:15
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An answer has been given by Michael Stoll at this MO-question, linked at this MSE-duplicate, using the elliptic curve
$$E_n \colon y^2 = x \bigl(x^2 + (4n(n+3)-3)x + 32(n+3)\bigr) =: x(x^2 + Ax + B),$$
where in our case $n=4$. Then the curve is known to have rank $1$, and thus there is a solution in positive integers of "truly enormous size", see the article "An Unusual Cubic Representation Problem" by Andrew Bremner. It is given by $$ x = 4373612677928697257861252602371390152816537558161613618621437993378423467772036; $$
$$ y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579; $$
$$ z = 154476802108746166441951315019919837485664325669565431700026634898253202035277999 $$ There are other solutions in positive integers, of course, but this is the smallest one.
answered Jun 14 '17 at 19:20
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2$\begingroup$ Typical of the expected enormity of whole solutions, when they exist, in some elliptic curves. $\endgroup$ – Piquito Jun 14 '17 at 19:26
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$\begingroup$ Is this solution possibly a multiple of Eric Lee's partly negative solution $(11,9,-5)$? $\endgroup$ – Hagen von Eitzen Jun 14 '17 at 19:56
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$\begingroup$ I have a negative number in my solution, I didn't realize they had to be positive $\endgroup$ – Eric Lee Jun 14 '17 at 21:22
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$x=11$, $y=9$, $z=-5$
I brute forced it.
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2$\begingroup$ The question has more detail than the title, and requires positive integer solutions. $\endgroup$ – user326210 Jun 14 '17 at 17:57
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2$\begingroup$ Still, it's good to know that there are any integer solutions at all. $\endgroup$ – user49640 Jun 14 '17 at 17:57
