<s>[INST] Show that $x^3+2y^3+4z^3=0 $ has no non trivial solutions without using infinite descent The question is to show that the equation $$x^3+2y^3+4z^3=0 $$ has no (non trivial) integer solutions. I know it can be done using infinite descent, but how do you do it using only the modular arithmetic argument?
 [/INST] Mod $9$ also solves it.
By Binomial Theorem $(3k\pm 1)^3\equiv \pm 1\pmod{9}$,
so $x^3\equiv\{0,\pm 1\}\pmod{9}$
(also $x^3\equiv \{0,\pm 1\}\pmod{8}$, so mod $8$ solves it similarly, as a comment has suggested).
$x^3+2y^3+4z^3\equiv 0\pmod{9}$
is equivalent to $x^3+2y^3\equiv -4z^3\pmod{9}$
$x^3+2y^2\equiv \{\pm 3,\pm 2, \pm 1, 0\}\pmod 9$
while $-4z^3\equiv \{0,\pm 4\}\pmod{9}$.
Therefore $x^3+2y^3\equiv -4z^3\equiv 0\pmod{9}$.
So $3\mid x,y,z$. Therefore $(x,y,z)$ is a solution if and only if $\left(\frac{x}{3},\frac{y}{3},\frac{z}{3}\right)$ is also a solution, so $(x,y,z)=(0,0,0)$ (see, e.g., proof by infinite descent). As a comment has said, you must use infinite descent because $x\equiv y\equiv z\equiv 0$ is a solution mod anything.
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