TITLE: Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.
QUESTION [7 upvotes]: Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.

This is a homework question, and I understand that its part of Fermat's Last Theorem, but when I looked that up to try to figure out the homework, I realized that it is way further than what we've learned in class so far. We have to prove this, but the only way I can think to do it is by exhaustion, which would be extremely lengthy. Any little shove in the right direction would be appreciated. Thanks!

REPLY [5 votes]: Here’s the stupidest possible way to do it: $10^3$ is only one thousand, and there are only nine cubes less than that. They are $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$ and $729$. Since all are less than $500$ except $512$ and $729$, you only need to check whether $488$ or $271$ is on the list, and neither is.