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twin primes  and  zygodrome

First numbers that belong to both sets:

11, 11777, 22111, 22277, 55333, 77711, 88811, 1115533, 2266633, 2277733, 2288833, 3311177, 3333311, 3335533, 3335599, 3338899, 3339977, 3366677, 3388999, 4441111, 4442233, 4446677, 5551111, 5552233, 5553311, 5566111, 5566333, 5588111, 6622999, 6662233, 6688111, 8800999, 8844499, 8885533, 9922111, 9922277, 9995599, 11000111, 11133377, 11144477, 11166611, 11177711, 11666111, 22233311, 22255999, 22888111, 33322111, 33322999, 33355577, 44000111, 44444999, 44488111, 44499977, 55500077, 55511999, 55555111, 66633377, 66644111, 77111999, 77333777, 77788811, 99955111, 99955577, 110022299, 110044999, 110099777, 111001199, 111111199, 111118877, 111333377, 111334499, 111336611, 111554477, 111557711, 111557777, 111660011, 111662233, 111667711, 111771199, 111772277, 111881111, 111888899, 112244477, 112255511, 112288333, 113332277, 114447733, 114447799, 115511111, 115544411, 115551133, 115551199, 116655599, 116665511, 117766333, 117775577, 117788833, 117788899, 118822999, 118883311, 220066699, 220088833, 220099333, 221117711, 221177711, 222006611, 222211333, 222277799, 222336677, 222667799, 222778877, 222881177, 222887711, 223330099, 223399333, 225533311, 225556633, 225557777, 226622777, 226662211, 227711777, 227766611, 228866777, 228885511, 229911133, 229988111, 229992211, 229993333, 229999111, 330088333, 331122299, 331155511, 331166111, 331166699, 331177333, 332255111, 333116699, 333311177, 333336611, 333443399, 333559999, 333661199, 333665599, 333771133, 333774433, 333990077, 333992299, 334442233, 335557711, 335559977, 335588833, 336622333, 336666611, 337773311, 338811199, 338899111, 338899999, 339922111, 339922211, 339988877, 440033777, 440066611, 441115511, 441133111, 442255999, 442277111, 442288111, 442288333, 443311333, 443377999, 444110099, 444113399, 444224411, 444338833, 444433111, 444433333, 444455111, 444466333, 444550033, 444660077, 444663377, 444777733, 444885599, 445557799, 445559911, 445559999, 445588111, 446677711, 447779911, 447779999, 447788777, 447788899, 448887799, 449911199, 449977111, 551122277, 553339933, 553366633, 553366999, 553377733, 554422277, 554455333, 555005533, 555008899, 555222299, 555566699, 555588811, 555661111, 555666611, 555883333, 555990077, 556622111, 557772211, 558844999, 559933333, 660033377, 661117799, 661133399, 662233111, 662299111, 663330011, 665551111, 665553377, 666003311, 666113311, 666116699, 666228811, 666229999, 666336677, 666644411, 666661111, 666770011, 666773311, 668800999, 668866111, 668866999, 669911111, 669966677, 771116611, 771117733, 772222277, 772223399, 772277111, 773334433, 774499933, 775550099, 775559999, 775588811, 777002299, 777007799, 777447733, 777661111, 777664499, 777666677, 777700333, 777993311, 778844777, 779911999, 880066699, 881119999, 882211199, 882211333, 882211999, 882224477, 882244333, 883333111, 883355111, 884477777, 884488333, 885544999, 885556699, 885557711, 885566699, 888110099, 888114499, 888449999, 888551177, 888811999, 888866611, 888991111.

Some larger examples :