Can M³+N³+ 4 be a perfect cube, whenever each of M and N is a positive integer?

If so, give an example.
If not, prove it.

It's just a question of picking a suitable mod.

The cubes, mod9 are worth {0,1,8}, so any two taken together are worth {0,1,2,7,8} mod 9.

Since the cubes mod9 are worth {0,1,8}, a number that's 4 less than a cube is worth {4,5,6} mod 9.

So the sum of two cubes can never be 4 less than a cube.

Or 5 less, in case anyone is thinking of asking. 

 

Edited on April 5, 2015, 9:50 pm

Posted by broll on 2015-04-05 21:16:00