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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 -

    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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