TITLE: Prove that there are no such different positive numbers that satisfy both $a+b=c+d$ and $a^3+b^3=c^3+d^3$.
QUESTION [8 upvotes]: $a,b,c$ and $d$ are all different and positive. Prove that it can't be true that equalities $a+b=c+d$ and $a^3+b^3=c^3+d^3$ are both correct at the same time. I sadly don't know how to start trying to solve this and haven't really tried much, so I'd like to get some ideas. Thanks.

REPLY [4 votes]: If we have $a+b=c+d$ with all different positive integers, we may as well assume $a$ is the smallest of them and write $c=a+r$ and $d=a+s$ with positive integers $r$ and $s$, in which case $b=a+r+s$.  A modicum of algebra shows that
$$(a^3+b^3)-(c^3+d^3)=6ars+3r^2s+3rs^2\gt0$$
so $a^3+b^3$ cannot equal $c^3+d^3$.