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de Polignac n.  and  nialpdrome

First numbers that belong to both sets:

1, 331, 877, 977, 997, 4311, 6433, 6521, 7331, 7431, 7753, 8111, 8411, 8621, 8831, 8861, 8873, 8887, 9431, 9521, 9641, 9941, 33221, 44111, 44431, 53321, 54311, 55331, 55421, 55511, 55553, 63221, 64211, 65111, 65531, 65533, 66221, 66551, 66663, 75521, 76441, 76631, 77443, 77771, 82111, 84221, 85411, 85421, 85541, 86411, 86441, 86521, 86641, 86653, 88421, 88511, 88543, 88631, 88633, 88643, 88711, 88841, 94421, 95411, 95441, 96411, 96431, 96553, 96611, 96643, 97211, 97421, 97631, 97633, 97661, 98533, 98765, 98773, 98821, 98831, 98861, 98863, 98887, 99311, 99541, 99853, 99983, 332111, 333221, 422111, 432221, 443111, 443321, 444221, 532211, 544321, 544421, 554111, 554443, 555221, 555521, 611111, 622211, 632111, 633221, 643111, 643211, 653221, 653333, 654433, 655531, 663111, 664211, 664331, 664441, 665441, 666221, 666521, 744221, 744311, 754441, 755531, 755533, 764311, 764411, 766211, 766553, 766631, 766661, 766663, 772111, 773311, 774433, 775543, 775553, 776411, 776443, 776533, 776661, 776665, 777221, 777431, 777541, 777551, 777641, 821111, 822221, 832211, 843221, 851111, 853331, 853333, 854333, 854441, 854443, 855521, 863221, 865211, 866321, 866441, 866553, 866651, 866653, 874421, 874433, 875531, 875533, 875543, 876221, 876431, 876551, 877541, 877543, 877661, 877733, 877753, 881111, 884411, 886111, 886663, 887111, 887321, 887555, 887741, 887743, 888551, 888643, 888833, 888881, 911111, 933311, 942211, 943331, 944441, 953311, 954433, 955333, 955421, 955511, 963211, 963311, 964333, 965533, 966521, 966641, 966643, 973211, 975311, 975433, 976421, 976543, 976663, 977531, 977621, 977651, 977741, 977743, 977771, 977773, 984433, 985321, 985541, 986333, 987431, 987433, 987551, 988331, 988541, 988633, 988751, 988753, 988843, 988871, 988873, 992111, 994211, 996311, 996643, 997333, 997541, 997721, 997733, 997751, 998111, 998321, 998411, 998533, 998663, 998711, 998743, 998771, 999411, 999611, 999665, 999721, 999821, 999851, 999853, 999931, 999973, 3311111, 4222111, 4222211, 4331111, 4421111, 4433221, 5111111, 5321111, 5332211, 5422211, 5433331, 5444111, 5444221, 5444441, 5532221, 5543211, 5543221, 5544331, 5544421, 5554433.

Some larger examples :