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Sir,
Going by your logic , there would not be any 'smallest number' .
E.g. -91 = (-3) ^3 + (-4) ^3 = (-6) ^3 + (5) ^3 < 91
and so the negative of any larger such positive numbers will give further smaller numbers (e.g. -1729) and so on ............
So I think this condition you are saying has to be implicitly implied :)