generalizations of the delone nagell equation

Question

Are there any references that study integer solutions to cubic Diophantine equations of the form $x^3 + 2y^3 = 2^a 3^b$ for $\{a,b\}\subset \{0,1,2,3\}$? I am aware that Nagell solved $x^3 + 2y^3 = 1$ $(x,y)=(1,0)$ is the only solution. I read (in the textbook Topics in Number Theory by Leveque, but no reference was provided) that Nagell proved that $x^3 + 2y^3 =3$ only has $(x,y) = (1,1)$ as its solution. Thanks in advance!

Answer 1

First of all, it is not true that $x^3 + 2y^3 =3$ only has $(1,1)$ as solution. Indeed, Nagell proved that the only solutions are $(1,1)$ and $(-5,4)$. The comments answer your question in some other cases. Actually, there is a more general case of your equation which is of interest and which has been studied, namely Thue equations. I will refer to the book:

Chaohua Jia, Matsumoto Kohji. Analytic Number Theory. Springer, Developments in Mathematics, 2002

In 1909, Thue proved that every equation of the form:

$$ p(x,y)=M $$

with $p$ irreducible polynomial in $\mathbb{Z}$ and $M \in \mathbb{Z}$ different from $0$, only has a finite number of integer solutions. In 1989, Tzanakis and de Weger gave an algorithm for finding bounds on $|x|$ and $|y|$ (see Tzanakis, N.; de Weger, B. M. M. On the Practical Solution of the Thue Equation. J. Number Th. 31, 99-132, 1989). There are some methods (discussed for instance in the book cited above) to find the maximal number of solutions in some particular cases. Your equation is generalised by:

$$ ax^3+by^3=c $$

Nagell and Delaunay proved that, if $b=c=1$, this equation has at most one solution with $xy \neq 0$. Then, the case with $c=1$ or $c=3$ was settled by Nagell, who proved that the equation has again at most one solution with $xy \neq 0 $ except when $a=1$, $b=2$ or $a=2$, $b=1$. More generally, if $p$ is a cubic homegeneous irreducible polynomial with negative discriminant $\Delta \neq -23, -31, -44$, then the equation with $M=1$ has at most three solutions (this was proved by Nagell and Delaunay). An explicit example is the following one ($a \geq 2$):

$$ x^3 + axy^2 + y^3 = 1 $$

It is easy to see that $(0,1)$, $(1,0)$ and $(1,-a)$ are solutions. By the above result, these are the unique ones.