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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 :)