math
posted by anonymous .
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^3b^3(ab)^3
=3ab(ab)
The likely candidates are:
990=3*11*5(115)... 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(ab).