Integer solutions for $a+b+c+d=0$, $a^3+b^3+c^3+d^3=24$

Question

How to get all integer solutions for $$a+b+c+d=0,$$ and $$a^3+b^3+c^3+d^3=24$$

So far I've put $a=-b-c-d$ into 2nd equation and try to factorise it, but didn't find anything useful.

Answer 1

Since $d=-a-b-c$, you can check the following identity:

$$a^3+b^3+c^3+d^3=-3(a+b)(a+c)(b+c).$$

Thus $$(a+b)(a+c)(b+c)=-8,$$

and that easily gives you all integer solutions. For instance, the case

\begin{cases} a+b=1 \\ a+c=1\\ b+c=-8 \end{cases}

gives you $(a,b,c,d)=(5,-4,-4,3),$ and the case

\begin{cases} a+b=-2 \\ a+c=-2\\ b+c=-2 \end{cases}

gives you $(a,b,c,d)=(-1,-1,-1,3).$