▁all ▁ 8 - tu ples . ▁The ▁first ▁term ▁in ▁these ▁sequences ▁corresponds ▁to ▁the ▁first ▁prime ▁in ▁the ▁smallest ▁prime ▁const ell ation ▁shown ▁below . ▁ ▁Ad miss ibility ▁In ▁order ▁for ▁a ▁k - tuple ▁to ▁have ▁infinitely ▁many ▁positions ▁at ▁which ▁all ▁of ▁its ▁values ▁are ▁prime , ▁there ▁cannot ▁exist ▁a ▁prime ▁p ▁such ▁that ▁the ▁tuple ▁includes ▁every ▁different ▁possible ▁value ▁mod ulo   p . ▁For , ▁if ▁such ▁a ▁prime ▁p ▁existed , ▁then ▁no ▁matter ▁which ▁value ▁of ▁n ▁was ▁chosen , ▁one ▁of ▁the ▁values ▁formed ▁by ▁adding ▁n ▁to ▁the ▁tuple ▁would ▁be ▁divis ible ▁by   p , ▁so ▁there ▁could ▁only ▁be ▁finit ely ▁many ▁prime ▁pla c ements ▁( only ▁those ▁including ▁p ▁itself ). ▁ ▁For ▁example , ▁the ▁numbers ▁in ▁a ▁k - tuple ▁cannot ▁take ▁on ▁all ▁three ▁values ▁ 0 , ▁ 1 , ▁and ▁ 2 ▁mod ulo ▁ 3 ; ▁otherwise ▁the ▁resulting ▁numbers ▁would ▁always ▁include ▁a ▁multiple ▁of ▁ 3 ▁and ▁therefore ▁could ▁not ▁all ▁be ▁prime ▁unless ▁one ▁of ▁the ▁numbers ▁is ▁ 3 ▁itself . ▁A ▁k - tuple ▁that ▁satisfies ▁this ▁condition ▁( i . e . ▁it ▁does ▁not ▁have ▁a ▁p ▁for ▁which ▁it ▁covers ▁all ▁the ▁different ▁values ▁mod ulo   p ) ▁is ▁called ▁adm iss ible . ▁ ▁It ▁is ▁conject ured ▁that ▁every ▁adm iss ible ▁k - tuple ▁matches ▁infinitely ▁many ▁positions ▁in ▁the ▁sequence ▁of ▁prime ▁numbers . ▁However