TITLE: A diophantine equation with only "titanic" solutions
QUESTION [27 upvotes]: I made a note some time ago that I had read in some book that the equation
$$313(x^3+y^3)=t^3$$
has positive integer solutions, but that these are so large that it would be absolutely hopeless to search for them with computers.  Unfortunately, I didn't write down where I read this and if you only have the equation, the results Google gives you aren't very helpful.  I could only find this so far.
Can someone point me to an article or book where I can read more about this equation?  (Preferably something with a proof of the claim above which is accessible even if you're not an expert in number theory.)

REPLY [13 votes]: The curve $a^3+b^3=N$ is birationally equivalent to the elliptic curve $y^2=x^3-432N^2$ with
\begin{equation}
a=\frac{36N+y}{6x} \hspace{2cm} b=\frac{36N-y}{6x}
\end{equation}
For $N=313^2$, the curve has rank $1$ and generator $G=(x_0,y_0)$ with
$x_0$=426235512202934545020503360093256801131707221692968586587468/216170759226021502298882345008844433022529079715666681
$y_0$=278275087731298331021683520315726613848790652329435004093249928083293904849586928211092140/100506794432879496007544646276171310440319758686599267034949687655666070652158579
which give the solution Noam Elkies tabulated.
The curve has no torsion points so all rational solutions come from points of the form $mP, \, \, m=1,2,\ldots$.
Looking at the above transformations, a positive solution will only arise when $|y| < 36N$ since $x>0$ always.
Experiments show that this first happens when $m=9$ giving a solution with roughly 6770 digits