let's see if you're as good as you claim to be @swissgirl

Prove that there are no positive perfect cubes less 1000 that are the sum of the cubes of two positive integers

Question

let's see if you're as good as you claim to be @swissgirl 

Prove that there are no positive perfect cubes less 1000 that are the sum of the cubes of two positive integers

lgbasallote · Answer

Note: no brute force substitution allowed

anonymous · Answer

can i use fermat?

lgbasallote · Answer

....isn't that like 30 pages long?

lgbasallote · Answer

.well according to urban myth anyway....

anonymous · Answer

actually the proof for the cases $n=4$ is not too bad, really like pythagoras
the proof for the case $n=3$ is also doable
the general case is well... you know that story study for 20 years and then come back and explain it to me

lgbasallote · Answer

oh. so the one that's 30 pages long is a^n + b^n = c^n...well then how does n = 3 look like?