, ▁there ▁is ▁no ▁adm iss ible ▁tuple ▁for ▁which ▁this ▁has ▁been ▁proven ▁except ▁the ▁ 1 - tuple ▁( 0 ). ▁Nevertheless , ▁by ▁Y it ang ▁Z hang ' s ▁famous ▁proof ▁of ▁ 2 0 1 3 ▁it ▁follows ▁that ▁there ▁exists ▁at ▁least ▁one ▁ 2 - tuple ▁which ▁matches ▁infinitely ▁many ▁positions ; ▁subsequent ▁work ▁showed ▁that ▁some ▁ 2 - tuple ▁exists ▁with ▁values ▁differ ing ▁by ▁ 2 4 6 ▁or ▁less ▁that ▁matches ▁infinitely ▁many ▁positions . ▁ ▁Pos itions ▁matched ▁by ▁in ad miss ible ▁patterns ▁Although ▁( 0 , ▁ 2 , ▁ 4 ) ▁is ▁not ▁adm iss ible ▁it ▁does ▁produce ▁the ▁single ▁set ▁of ▁pr imes , ▁( 3 , ▁ 5 , ▁ 7 ). ▁ ▁Some ▁in ad miss ible ▁k - tu ples ▁have ▁more ▁than ▁one ▁all - prime ▁solution . ▁This ▁cannot ▁happen ▁for ▁a ▁k - tuple ▁that ▁includes ▁all ▁values ▁mod ulo   3 , ▁so ▁to ▁have ▁this ▁property ▁a ▁k - tuple ▁must ▁cover ▁all ▁values ▁mod ulo ▁a ▁larger ▁prime , ▁imp lying ▁that ▁there ▁are ▁at ▁least ▁five ▁numbers ▁in ▁the ▁tuple . ▁The ▁short est ▁in ad miss ible ▁tuple ▁with ▁more ▁than ▁one ▁solution ▁is ▁the ▁ 5 - tuple ▁( 0 , ▁ 2 , ▁ 8 , ▁ 1 4 , ▁ 2 6 ), ▁which ▁has ▁two ▁solutions : ▁ ▁( 3 , ▁ 5 , ▁ 1 1 , ▁ 1 7