▁the ▁overhead ▁of ▁prim ality ▁testing ▁to ▁obtain ▁the ▁prime ▁numbers ▁as ▁candidate ▁factors . ▁A ▁useful ▁table ▁need ▁not ▁be ▁large : ▁P ( 3 5 1 2 ) ▁= ▁ 3 2 7 4 9 , ▁the ▁last ▁prime ▁that ▁fits ▁into ▁a ▁six teen - bit ▁signed ▁integer ▁and ▁P ( 6 5 4 2 ) ▁= ▁ 6 5 5 2 1 ▁for ▁unsigned ▁six teen - bit ▁integers . ▁That ▁would ▁suff ice ▁to ▁test ▁prim ality ▁for ▁numbers ▁up ▁to ▁ 6 5 5 3 7 2 ▁= ▁ 4 , 2 9 5 , 0 9 8 , 3 6 9 . ▁Pre par ing ▁such ▁a ▁table ▁( us ually ▁via ▁the ▁S ieve ▁of ▁Er at ost hen es ) ▁would ▁only ▁be ▁worth while ▁if ▁many ▁numbers ▁were ▁to ▁be ▁tested . ▁If ▁instead ▁a ▁variant ▁is ▁used ▁without ▁prim ality ▁testing , ▁but ▁simply ▁div iding ▁by ▁every ▁odd ▁number ▁less ▁than ▁the ▁square ▁root ▁the ▁base - 2 ▁n ▁digit ▁number ▁a , ▁prime ▁or ▁not , ▁it ▁can ▁take ▁up ▁to ▁about : ▁ ▁In ▁both ▁cases , ▁the ▁required ▁time ▁grows ▁expon entially ▁with ▁the ▁digits ▁of ▁the ▁number . ▁ ▁Even ▁so , ▁this ▁is ▁a ▁quite ▁satisf actory ▁method , ▁considering ▁that ▁even ▁the ▁best - known ▁algorithms ▁have ▁exponential ▁time ▁growth . ▁For ▁a ▁chosen ▁uniformly ▁at ▁random ▁from ▁integers ▁of ▁a ▁given ▁length , ▁there ▁is ▁a ▁ 5 0 % ▁chance ▁that ▁ 2