(This post was motivated by an old one.) For Pell equations,
$$x^2-dy^2 = 1\tag{1}$$
and $d<100$, the largest fundamental solution is for $d = 61$ (which happens to be the 6th power of a fundamental unit),
$$x+y\sqrt{61} = 1766319049+226153980\sqrt{61} =\left(\frac{39+5\sqrt{61}}{2}\right)^6$$
In general, for prime $d = 8n+5$ and odd fundamental solution to $u^2-dv^2=-4$, then initial solution to $(1)$ will be a 6th power and thus huge. For the cubic analogue,
$$x^3+dy^3+d^2z^3-3dxyz = 1\tag{2}$$
just like the Pell, from an initial solution, an infinite more can be found. But how do we characterize "tricky" $d$ such that the smallest positive $x,y,z$ of $(2)$ are in fact relatively large. (From limited data, I think prime $d = 12n-1$ is one subset).
Question: Anybody knows how to find $d=23,47,59,71$ of $(2)$?
(P.S.1 It would suffice to find "small" signed $x,y,z$ since one can derive the positive ones from those. For example, for $d=11$ and $x,y,z = 1,4,-2$, one can derive $x,y,z = 89, 40, 18$.
(P.S.2 Useful details can be found in Springer's The Cubic Analogue of Pell's Equation.)
The question was to solve,
$$x^3+dy^3+d^2z^3-3dxyz = 1\tag{1}$$
for $d = 23, 47, 59, 71$. (Note: The equation $(1)$ can be solved in the integers for all non-cube integer $d$.) Seiji Tomita has done more and created a table of fundamental solutions for non-cube $d<100$. The largest "smallest" $x,y,z$ found so far is for $d = 69 = 3\cdot23$,
$$x,y,z ={13753611475894008059401,\;-5630668308465438120720,\; 555253697459615284770}$$
From this, one can derive the "smallest" positive solution to $(1)$ for $d = 69$ (and from which an infinite more can then be found),
$$x,y,z = 404886837053487091694212951195653956127452401,\; 98715184393700556938337454013404500951638820,\; 24067681974543893805323831567684099602695630$$
Analogous to Pell equations, the $x/y, y/z$ are close to $69^{1/3}$ (within $10^{-68},10^{-67}$, respectively). The ratios of larger positive $x,y,z$ will get ever closer and closer to $d^{1/3}$.
P.S. Pell equations appear in a lot of contexts, and can solve other Diophantine equations, like $x_1^3+x_2^3+x_3^3=1$. Right now $(1)$ seems to be only a mathematical curiosity, but maybe someday, someone, somewhere can use it to prove that such-and-such equation has an infinite number of integer solutions.