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.
Integer solutions for $a+b+c+d=0$, $a^3+b^3+c^3+d^3=24$
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diophantine-equations
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1Trivial solution includes all permutations of $(3,-1,-1,-1) $. – 2017-01-07
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1@Rohan: Why do you refer to this as *trivial* (i.e., what makes it more trivial than any other solution)? – 2017-01-07
1 Answers
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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 $(a,b,c,d)=(-1,-1,-1,3).$