What is the largest integer less than 1000 that can be represented as a^3+b^3+c^3 for some integers a,b,c, such that a+b+c=0.
By trial and error, I get:
11^3+(-5)^3+(-6)^3=990
If we assume the largest integer is N such that
N(a,b)=a^3-b^3-(a-b)^3
=3ab(a-b)
The likely candidates are:
990=3*11*5(11-5)... a,b,c = 11, -5, -6
993=3*331
996=3*332=3(4*83)
999=3*333=3(9*37)
All of which cannot be reduced to the form
3ab(a-b).