TITLE: Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?
QUESTION [9 upvotes]: Does the equation  $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions?  This is equivalent to a conjecture about OEIS sequence A245624.
Maple tells me this is a curve of genus $1$, and its Weierstrass form is $s^3 + t^2 + 20 = 0$, with $$ \eqalign{ s = \dfrac{-2(7 y^2 + 6 y + 2)}{x^2},& \
t = \dfrac{-2(3 x^3 + 14 y^2 + 12 y + 4)}{x^3}\cr
x = \dfrac{-2s(t-6)}{s^3+56},&\ y = \dfrac{4t-24}{s^3+56}}$$
So I can find rational points on both curves, but I haven't been able to find integer points on (1) other than the trivial $(0,0)$.

REPLY [3 votes]: For what it's worth, I took a generator $P$ of the Mordell-Weil group of $y^2=x^3-20$ and computed the preimage of $k\cdot P$ for $|k| \leq 300$ back to the original curve, and the only instance where the point was integral was for $k=0$.