Solve the diophantine equation $x^3-y^3-z^3=3^t \cdot 2xyz,(x,y,z \in \Bbb N)$, where $t\in \Bbb N.$
We can find infinitely many solutions for $t=1$ from initial solution $x=52,y=21,z=19$.
I search for all $x<13000$ but find no solution for $t>1$, I want to know if there are any solutions for $t>1$.
Edit 1: Just noticed that the solution asks for $u,v,w\in\mathbb N$, so while the search procedure is the same we require an extra check that they are $\geq 1$.
Some solutions for small $t$'s.
\begin{array}{|c|c|c|c|} \hline t& u & v & w \\ \hline 2 & 35917476 & 6829645 & 10182731\\ \hline 3 & 1137565 & 647349 & 30196 \\ \hline 4 & 44334184670964 & 613899299195 & 18359866789309 \\ \hline \end{array} It appears that the torsion points correspond to a solution where one of $u,v,w$ is $0$, so we do not use them. Assuming this, solution to the original problem would imply finding points of infinite orders for an associated Elliptic curve.
for $t=5$ the curve is birational to an Elliptic curve of rank $0$ so there are no non-zero solutions. Hence not all $t$ yields a solution.
For $t\geq 6$ the curve parameters are huge enough that both Magma and Sage is unable to compute the generators. (Perhaps finding a solution would imply a better algorithm for finding generators?)
In particular at $t=6$ the original curve $$ \frac{46656}{114791255}(u^3 - v^3 - w^3 - 3^6 \cdot 2uvw) = 0 $$ is birational to $$ y^2z = x^3 - 1506290966208 x z^2 + 711559628544550032 z^3 $$ via $$ \begin{align} (u,v,w) &= (\frac{-1458 x+y+1033121268 z}{216}, \frac{1458 x+y-1033121268 z}{216} , \frac{-x-2125764 z}{36}) \end{align} $$
We can systematically try to solve for each $t>1$ and it seems like sometimes the equation will be birational to a rank 1 elliptic curve, would would mean infinitely many points.
Edit 1: Torsion points seems to corresponds to solutions where $u,v$ or $w=0$.
On the other hand, for each $1\leq t\leq 20$, the resulting Elliptic curve each have two torsion points, would in turn means two solutions in the original equation. It looks like this may hold for each $t$, so at least two (coprime) solutions for each $t$.$t=2$ example
For $t=2$, we want integer solutions to $$ f(u,v,w) := u^3 - v^3 - w^3 - 18uvw=0 $$ We may freely multiply a rational $t\neq 0$ to get $$ t\cdot f(u,v,w) = t(u^3 - v^3 - w^3 - 18uvw)=0 $$ Since scaling of $(u,v,w)$ to $(ku,kv,kw)$ also leads to a solution, it suffices to find any rational solution (then scaling it to integers if necessary).
Let $$ \begin{align} t&=-\frac{2^6\cdot 3^6}{5\cdot 43}=-\frac{46656}{215}\\ u&= \frac{-18x+y+1908z}{216}, & v&= \frac{18 x + y - 1908 z}{216}, & w &= \frac{-x-324}{216}, \end{align} $$ then it may be verified that this transform the original equation to $$ g(x,y,z):=x^3 - y^2 z - 36288 x z^2 + 2285712 z^3=0 $$ which is a projective elliptic curve. (In practice this was found using Sagemath's EllipticCurve_from_cubic() function.)
Due to the scaling, it suffices to set $z=1$ and find rational solutions $(x,y,1)$ instead, i.e. solving $$ E: y^2 = x^3-36288x+2285712 $$ This is a rank 1 Elliptic curve, so there are infinitely many rational points $(x,y,1)$. Each of these leads to a rational solution for $f(u,v,w)=0$, so we know that there are infinitely many solutions.
Sample solutions.
The curve $E$ has rank 1 with generator $G=(-168,1908)$, so that it generates points $$ \{(-168,1908), (\frac{1395408}{2809} , -\frac{1538896356}{148877}), (\frac{7075793089}{672624225} , \frac{24078086348446313}{17444509275375})\cdots\} $$ Converting to $(u,v,w)$, we get $$ \{(\frac{95}{3}, -14, -\frac{13}{3})),(-\frac{11972492}{148877},-\frac{6829645}{446631},-\frac{192127}{8427}),(\frac{54059017558123943}{3768014003481000},-\frac{5902844861231317}{3768014003481000},-\frac{225006041989}{24214472100})\cdots \} $$ Normalizing, the first three solutions are $$ \begin{align} (95,-42,-13),\\ (-35917476, -6829645, -10182731)\\ (54059017558123943, -5902844861231317, -35013190193908290) \end{align} $$
Edit 1: We require solutions in $\mathbb N$, so we look for positive triplets. Since we can multiply by $-1$, negative triplets work as well. Here we find the solution $$ (35917476, 6829645, 10182731) $$