<s>[INST] $x^3 +y^2 +z =100z+10y+x$ What is the largest and smallest integer that satisfies this equation. $x^3 + y^2 +z=\overline{zyx}$, where $\overline{zyx}$ denotes the sequence of the digits.
$x^3 +y^2 +z =100z+10y+x$,where $x,y,z>0$
The maximum value of $x$, $y$, $z$ individually can only be $9$.
$\text{Maximum value}= 9^3 + 9^2 + 9 = 819$
So, $100z+10y+z<819$
Then I'm lost here a bit. What do I do next?
 [/INST] $$x^3+y^2+z=100z+10y+x$$
$$x(x+1)(x-1)=y(10-y)+99z$$
No needs to check $x=1$ and $y$ have symmetrical sense (e.g. if $y=1$ satisfies so as $y=9$).
Also, $9\leq y(10-y) \leq 25$ hence $$9(1+11z)\leq x(x+1)(x-1) \leq 25+99z$$
$$108\leq x(x+1)(x-1) \leq 916$$
$$\because \quad 3\times 4\times 5 < 108 < 4\times 5\times 6 \implies x\geq 5$$
Now,
$$4\times 5\times 6 \leq y(10-y)+99z \leq 8\times 9\times 10$$
$$120-25 \leq 99z \leq 720-9 \implies 1\leq z \leq 7$$

This can further confine to $5\leq x \leq 8$; and in turn, furthermore confine to $1\leq z\leq 5$.

Simple search gives
$$\overline{zyx}=135,\, 175,\, 518,\, 598$$
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