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

The perfect cubes less than 1000 are: 

1^3 = 1 
2^3 = 8 
3^3 = 27 
4^3 = 64 
5^3 = 125 
6^3 = 216 
7^3 = 343 
8^3 = 512 
9^3 = 729 

so 
2^3 = 8 could only be 1 + 1, but this is 2, not 8. 
3^3 = 27 could be 1 + 1, 1 + 8 or 8 + 8, but all of these are too small 
4^3 = 64 could be 1 + 1, 1 + 8, 1 + 27, 8 + 8, 8 + 27, 27 + 27.
 general proof that a^3 + b^3 = c^3 holds for no positive integers 
hope it helps