<s>[INST] Show that there do not exist any distinct natural numbers a,b,c,d such that 
Show that there do not exist any distinct natural numbers a,b,c,d such that $a^3+b^3=c^3+d^3$ and $a+b=c+d$.

 [/INST] Mark $a+b=x$ then $b=x-a$ and $d=x-c$. Notice that $x\ne 0$. No we have: $$a^3+(x-a)^3 = c^3+(x-c)^3$$ and thus $$-3x^2a+3xa^2 = -3x^2c+3xc^2$$
so \begin{align}xa-a^2= xc-c^2& \implies x(a-c) = (a-c)(a+c)\\ &\implies x=a+c \\& \implies a+c=a+b \\&\implies c=b\end{align}
A contradiction.
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