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!
Asked
Active
Viewed 1,523 times
0
Trevor Gunn
- 27,041
-
Do you have the answer? – CY Aries Jun 14 '17 at 17:22
-
2what have ___you___ tried so far? – Dando18 Jun 14 '17 at 17:22
-
also what leads you to believe there is a solution? brute force shows no solutions for $x,y,z \le 250$ – Dando18 Jun 14 '17 at 17:52
-
I right now have it worked down to the form ${{x^3+y^3+z^3+xyz}\over{x^2y +x^2z+2xyz+xy^2+xz^2+y^2z+yz^2}} {+1=4}$ – Bensstats Jun 14 '17 at 18:02
-
I know that there is a solution because I know it. The tag of the question is puzzle. – soccerStack Jun 14 '17 at 18:10
-
2@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. – Χpẘ Jun 14 '17 at 18:20
-
@soccerStac: Do you really have in hand a solution with integer positive numbers? Wolfram gives the solution $(x,y,z)=(11,9,-5)$ from which there are trivially six solutions but with one of coordinates negative. Answer being sure to this, please. – Piquito Jun 14 '17 at 18:45
-
2Possible duplicate of Find answer of $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$ – Dietrich Burde Jun 14 '17 at 18:54
-
@DietrichBurde There is no solution there – soccerStack Jun 14 '17 at 19:11
-
1@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". – Dietrich Burde Jun 14 '17 at 19:15
1 Answers
6
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.
Dietrich Burde
- 130,978
-
2Typical of the expected enormity of whole solutions, when they exist, in some elliptic curves. – Piquito Jun 14 '17 at 19:26
-
Is this solution possibly a multiple of Eric Lee's partly negative solution $(11,9,-5)$? – Hagen von Eitzen Jun 14 '17 at 19:56