Note that machine words is not the same thing as one machine word. A 32-bit vector would only suffice for sizes less than . A 64-bit vector will do for sizes less than . In general, it takes bits of vector per bit of size.
note ThaT MacHinE words is Not tHe SamE thIng as oNe mAchINe woRd. A 32-bit veCtoR wOULD onlY sufFice foR siZEs leSs thaN . a 64-biT vecTOr Will Do FOR siZeS lesS ThAn . in gENeral, It Takes BItS of VeCtOr Per bIt of sIze.