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From research completely unrelated to Number Theory I stumbled onto the following equation:

$$ xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3 $$

for $x, y, z$ integers, $x,y,z \neq 0$. Are there nonvanishing integers that satisfy it (there are many solutions if one of them is zero)?

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    $\begingroup$ Minor comment: if you put $x=-w/2$ then the equation becomes $wyz=7((w+y+z)/6)^3$, which looks a bit nicer. $\endgroup$
    – Michael Zieve
    Commented May 14, 2014 at 11:49

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No. This can be verified via the following Magma code, which can be used on the free Magma online calculator:

P<x,y,z>:=ProjectiveSpace(Rationals(),2);
C:=Curve(P,x*y*z-7/16*((2*x-y-z)/3)^3);
E:=EllipticCurve(C);
MordellWeilGroup(E);

which outputs

Abelian Group isomorphic to Z/3
Defined on 1 generator
Relations:
    3*$.1 = 0

This shows that, when the given equation is viewed as a curve in $\mathbf{P}^2$, it has exactly three rational points. One can use Magma to automatically compute these three points, or just observe that there are obviously three rational points with $xyz=0$.

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  • $\begingroup$ I suppose a proof "by hand" is too much to ask for? $\endgroup$
    – Gerry Myerson
    Commented May 14, 2014 at 7:33
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    $\begingroup$ You can write the equation in Weierstrass form and use the point of order 3 to do a 3-descent. Since there are only three bad primes (2,3,7) this seems doable. (This works only if there are no elements of order 3 in the Tate-Shafarevich group). $\endgroup$
    – Remke Kloosterman
    Commented May 14, 2014 at 9:52
  • $\begingroup$ probably working in Sage will give you more info. It can identify the curve in certain database, etc... $\endgroup$
    – Dima Pasechnik
    Commented May 14, 2014 at 10:39
  • $\begingroup$ Thanks! That's fine enough by me although I agree that a proof by hand would be also nice. $\endgroup$
    – user22139
    Commented May 14, 2014 at 17:56
  • $\begingroup$ Michael, in case you noticed my earlier comments, I was able to put a little subset of my package in the Magma online calculator and get correct genera and spinor genera for two well-known positive ternary quadratic forms. This is great. They come out as symmetric (integer) matrices, i have separate programs to put them in the reduced form i prefer. $\endgroup$
    – Will Jagy
    Commented May 16, 2014 at 0:38

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