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.

Note: This is not a complete solution. It may give others some clues.
We have that $$(a-b)(a^2+b^2+ab+a+b+1)=8b^3 \tag{1}$$ Let 
$$d=\gcd(a-b, a^2+b^2+ab+a+b+1).$$
And let $p$ be an odd prime divisor of $d$. Then $p \mid a-b$ and $p \mid b$. It follows that $p|a$. Since $p \mid a^2+b^2+ab+a+b+1$ we get $p \mid 1$. Contradiction. Thus only prime divisor of $d$ is $2$.
From equation $(1)$ observe that $a$ and $b$ have the same parity. Thus $a-b$ is even. So let $a-b=c=2^k (2l+1)$, where $l$ is an integer. Equation $(1)$ becomes.
$$c(3b^2+c^2+3bc+2b+c+1)=8b^3 \tag{2}$$
Notice that if $b$ is an even number then $3b^2+c^2+3bc+2b+c+1$ will be odd. Hence 
$$d=\gcd\big(c, 3b^2+c^2+3bc+2b+c+1\big)=1.$$ And $c$ is a perfect cube. Now, suppose that $b$ is odd. Then, since $3b^2+1 \equiv 0 \pmod{4}$ we get
$$3b^2+c^2+3bc+2b+c+1=3b^2+1+c(c+1+3b)+2b \\ =4m+2b=2(2m+b)$$Thus the highest power of $2$ dividing LHS of equation $(2)$ is $k+1$. And since $v_2($RHS$)=3$ we get that $k=2$. Thus $a-b=4(2l+1)$ cannot be a cube.
Maybe in this case there is no solution for the equation at all.
