<s>[INST] If $a^3+a^2+a=9b^3+b^2+b$ and $a,b$ are integers then show $a-b$ is a perfect cube. If $a^3+a^2+a=9b^3+b^2+b$ and $a,b$ are integers then show $a-b$ is a perfect cube.
My attempt:I factorized it like below:
$(a-b)(a^2+b^2+ab+a+b+1)=8b^3=(2b)^3$
I take $gcd(a-b,a^2+b^2+ab+a+b+1)=d$
If $d=1$ then it is clear that $a-b$ is a perfect cube then consider $d>1$ then there is a $p$ that is prime and $p \mid d$.We have :
$p\mid a-b \Rightarrow p \mid (2b)^3 \Rightarrow p \mid 2b \Rightarrow p\mid 2$ or $p\mid b$
If $p\mid b$ then also $p\mid a$ (as $p\mid a-b$ holds). Then we will get to $p \mid 1$ because:
$ p \mid a^2+b^2+ab+a+b,p \mid a^2+b^2+ab+a+b+1 \Rightarrow p \mid 1$
which is clearly wrong then we have $p\mid 2$ so $p=2$ means $d=2^k$ where $k$ is a natural number including $0$.In the case $d=1$ we have the right result.So assemble $k \ge 1$.Because $2 \mid a-b$ we can conclude that $a,b$ have the same parity @Ghartal showed in his answer that if $a,b$ are both even we have a right result but if $a,b$ are both odd we don't.So maybe we have to prove $a,b$ can,t be both odd.
 [/INST] It turns out that there are only two rational solutions of
$a^3+a^2+a = 9b^3+b^2+b$.  One is the obvious integer solution
$(a,b)=(0,0)$, for which $a-b$ is the cube $0$.
The other is $(a,b) = (-7/5, -3/5)$, for which $a-b = -4/5$
is not a cube but $a,b$ are not integers.  So $a-b$ is indeed
a cube for all the integer solutions, as desired.
The proof that there are no other rational solutions is elementary,
but requires Fermat's method of descent.  The equation
$a^3+a^2+a = 9b^3+b^2+b$ defines a smooth cubic curve with
a rational point at $(a,b)=(0,0)$, so it's an elliptic curve
that we can bring to standard form $y^2=\text{cubic}(x)$ by
projecting from the origin.
Let $(a,b)$ be any rational solution other than $(0,0)$.
Then $b\neq 0$ because the nonzero solutions of $a^3+a^2+a$
are not rational (or even real). Write $a=mb$ and divide by $b$ to get
$$
(9-m^3) b^2 + (1-m^2) b + (1-m) = 0.
$$
This equation is quadratic in $b$, so it has rational solutions iff
the discriminant is a square.  The discriminant is
$$
-3 m^4 + 4 m^3 - 2 m^2 + 36 m - 35 = (1-m)(3m-7)(m^2+2m+5).
$$
Taking $m=1$ yields $8b^2 = 0$, so we're back to the $b=0$ solution;
and $m=7/3$ yields $-4(5b+3)^2/27 = 0$, so $b=-3/5$ and $a=-7/5$.
We claim that there are no other solutions.
If $m\neq 1$ we may write $m = 1+(8/x)$ and calculate
$$
-3 m^4 + 4 m^3 - 2 m^2 + 36 m - 35 = \frac{2^8}{x^4} (x^3-2x^2-16x-48),
$$
so we seek rational points on the elliptic curve
$$
y^2 = x^3-2x^2-16x-48 = (x-6)(x^2+4x+8)
$$
other than the "point at infinity".
The point $(x,y)=(6,0)$ brings us back to $m = 1 + (8/6) = 7/3$,
and it turns out there are no others.  Since the cubic on the right-hand side 
has a linear factor, we can try to prove this using Fermat's descent,
which turns out to be enough.  But these days there are other
resources we can try first: Cremona's mwrank or the
LMFDB, and either of these
tells us that there are no other solutions and that Fermat-style descent
suffices to prove it.  In the case of mwrank we'd need a bit of
additional computation to exclude other torsion points, so let's go the
database route: enter the coefficients [0,-2,0,-16,-48] into the
"curve, label or isogeny class label" search window for
elliptic curves
over $\bf Q$, and find the entry for
this curve of
conductor 544 (which is small enough that it's also in Cremona's book).
We find that the curve has only the known two torsion points
and rank zero, and moreover that the 2-part of the Tate-Shafarevich group
is trivial, so a 2-descent suffices to establish rank zero.
This completes the proof.
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