Prove there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.

Question

I am familiar with Fermat's Last Theorem that there are no integers such that $x^3+y^3=z^3$, but I need a simpler proof that demonstrates that fact when we know that $x$ and $y$ are positive, and $z=10^3$.

Thank you!

Answer 1

Write the positive cube numbers up to $10$:

$1^3=1,\quad2^3=8,\quad 3^3=27,\quad4^3=64,\quad5^3=125,\quad6^3=216,\quad7^3=343,\quad8^3=512,\quad9^3=729,\quad10^3 =1000$

Now all you need to do is demonstrate that the sum of any two combinations of these does not equal $1000$.