Posted by **anonymous** on Tuesday, July 2, 2013 at 12:58pm.

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.

- math -
**MathMate**, Tuesday, July 2, 2013 at 4:39pm
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).

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