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 :)

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 :)