Consider the seven positive integers: 123, 1234, 12345, 123456, 1234567, 12345678 and 123456789.
How many of these seven numbers are expressible as the difference of cubes of two positive integers?
All cubes are 0, 1, or 8 mod 9. The difference of two cubes has to be 0, 1, 2, 7, or 8 mod 9.
123=6 mod 9
1234=1 mod 9
12345=6 mod 9
123456=3 mod 9
1234567=1 mod 9
12345678=0 mod 9
123456789=0 mod 9
Now, we know that it has to be 1234, 1234567, 12345678, or 123456789. The answer is 1234567=642^3641^3.

Posted by Math Man
on 20141220 20:39:08 