Find all integers $a$ and $b$ such that $$a^3 \equiv b^3 \pmod{7^3}\,,$$ knowing that $7 \nmid ab$.
As a try, I noticed that, since $\gcd(b, 7)=1$, there exists $x \in \mathbb{N}$ such that $b\cdot x \equiv 1 \pmod{7} \Rightarrow b^3 \cdot x^3 \equiv 1 \pmod{7^3}$. Thus $a^3\equiv b^3 \pmod{7^3} \iff (ax)^3\equiv 1\pmod{7^3}$. After this I tried to use Euler's totient function, but I do not know where I should begin.
On
Given $7\nmid ab$, $a^3\equiv b^3\bmod 7^3\iff (a/b)^3\equiv1\bmod 7^3$.
Let $x\equiv a/b\bmod 7^3$. We are looking for $x$ such that $7^3|x^3-1=(x-1)(x^2+x+1)$.
Now if $7|x-1$ and $x^2+x+1$, then $7|x^2+x+1-(x+2)(x-1)=3$, a contradiction.
So $7^3|x-1$ (i.e., $x\equiv1\bmod7^3$) or $7^3|x^2+x+1$.
Now we are looking for $x$ such that $x^2+x+1\equiv0\bmod7^3$.
Note that $x\equiv 2$ and $x\equiv 4$ are the solutions to $x^2+x+1\equiv0\bmod7$.
If $x=7k+4$ is a solution to $x^2+x+1\equiv0\bmod7^2$, then $k\equiv2\bmod7$, so $x\equiv18\bmod7^2$.
If $x=7^2k+18$ is a solution to $x^2+x+1\equiv0\bmod7^3$,
then $k\equiv0\bmod7$, so $x\equiv18\bmod7^3$.
If $x=7k+2$ is a solution to $x^2+x+1\equiv0\bmod7^2$, then $k\equiv4\bmod7$, so $x\equiv30\bmod7^2$.
If $x=7^2k+30$ is a solution to $x^2+x+1\equiv0\bmod7^3$,
then $k\equiv6\bmod7$, so $x\equiv324\bmod7^3$.
(I could have argued that, if $x\equiv18$ is a solution,
then $x^4+x^2+1\equiv x+x^2+1\equiv0$, so $x^2\equiv18^2=324$ is a solution too.)
So either $a\equiv b$ or $a\equiv18b$ or $a\equiv324 b\bmod 7^3$, and then $a^3\equiv b^3\bmod 7^3$.
For integers $a$ and $b$, $a^3\equiv b^3\pmod{7^3}$ if and only if $$(a-b)\,(a^2+ab+b^2)\equiv 0\pmod{7^3}\,.$$ If $7\mid a-b$ and $7\mid a^2+ab+b^2$, then it follows that $$3ab=(a^2+ab+b^2)-(a-b)^2\equiv 0\pmod{7}\,,$$ whence $7\mid 3ab$, so $7\mid ab$. This means $7\mid a$ or $7\mid b$. However, as $7\mid a-b$, we conclude that $7\mid a$ and $7\mid b$. Thus, $a^3\equiv 0\equiv b^3\pmod{7^3}$ in this case. From now on, we assume that $7\nmid a-b$ or $7\nmid a^2+ab+b^2$.
In the case $7\nmid a^2+ab+b^2$, we conclude that $7^3\mid a-b$. Therefore, $a\equiv b\pmod{7^3}$, but $7\nmid a$ and $7\nmid b$. The only remaining case is when $7\nmid a-b$.
When $7\nmid a-b$, we get $7^3\mid a^2+ab+b^2$. Thus, $7\mid a^2+ab+b^2$. Now, observe that $$a^2+ab+b^2\equiv (a-2b)(a-4b)\pmod{7}\,.$$ That is, $a\equiv 2b\pmod{7}$ or $a\equiv 4b\pmod{7}$.
Suppose that $a\equiv 2b\pmod{7}$. Then, $a-2b=7k$ for some integer $k$. Therefore, $$a^2+ab+b^2=(2b+7k)^2+(2b+7k)b+b^2=7\left(b^2+5kb+7k^2\right)\,.$$ Because $7^3\mid a^2+ab+b^2$, we deduce that $$7^2\mid b^2+5kb+7k^2\,.$$ Therefore, $$b(b-2k)\equiv b^2+5kb+7k^2\equiv 0\pmod{7}\,.$$ However, we can easily see that $b\not\equiv 0\pmod{7}$. This implies $$b\equiv 2k\pmod{7}\,.$$ Write $b-2k=7l$ for some integer $l$. Then, $$b^2+5kb+7k^2=(2k+7l)^2+5k(2k+7l)+7k^2=7(3k^2+9kl+7l^2)\,.$$ Thus, we need $7\mid 3k^2+9kl+7l^2$. Consequently, $$3k(k+3l)\equiv 3k^2+9kl+7l^2\equiv 0\pmod{7}\,.$$ Because $b-2k=7l$ and $7\nmid b$, we conclude that $7\nmid k$, whence $$k\equiv -3l\pmod{7}\,.$$ Therefore, $k=-3l+7m$ for some integer $m$, whence $$b=2k+7l=(-3l+7m)+7l=l+14m$$ and $$a=2b+7k=2(l+7m)+7(-3l+7m)=-19l+77m\,.$$ Thus, $$a=-19(b-14m)+77m=-19b+343m\equiv -19b\pmod{7^3}\,.$$
Now, suppose that $a\equiv 4b\pmod{7}$. Then, $a-4b=7k$ for some integer $k$. Therefore, $$a^2+ab+b^2=7(3b^2+9kb+7k^2)\,.$$ Again, we can see that $$3b(b+3k)\equiv 3b^2+8kb+7k^2\equiv 0\pmod{7}\,.$$ Ergo, $b\equiv -3k\pmod{7}$. Let $b=-3k+7l$ for some integer $l$. Then, $$3b^2+9kb+7k^2=7(k^2-9kl+21l^2)\,,$$ whence $$k(k-2l)\equiv k^2-9kl+21l^2\pmod{7}\,.$$ Thus, $k\equiv 2l\pmod{7}$. Write $k=2l+7m$. Therefore, $$b=-3k+7l=l-21m$$ and $$a=4b+7k=4(l-21m)+7(2l+7m)=18l-35m\,.$$ Finally, we may write $$a=18(b+21m)-35m=18b+343m\equiv 18b\pmod{7^3}\,.$$
Remark. In principle, there should be a more concise solution using Eisenstein integers. I just opted for an elementary solution.