Problems & Puzzles: Puzzles

 Puzzle 1042. Another puzzle about emirps J. M. Bergot sent the following nice puzzle. 709, 739, and 769 are emirps such that 907-709= 937-739=967-769=k=198. Q. Find larger sets of emirps with the above property: same k value.

During the week from 19 to 25 of June, 2021, Contributions came from Emmanuel Vantieghem, Simon Cavegn, Giorgios  Kalogeropoulos, Seiji Tomita, Adam Srtinchcombe, Hakan Summakoglu, Metin Sariyar, Oscar Volpatti.

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Emmanuel wrote:

The maximum number of emirps < 10^10  with the same  k-value I could find was  1242, with  k-value  9999990.
Here are these emirps :
100131011, 100151011, 100292011, 100373011, 100575011, 101060111, 101090111, 101414111, 101474111, 101858111, 101929111, 102121211, 102171211, 102303211, 102616211, 102636211, 103919311, 104444411, 105232511, 105636511, 105747511, 106212611, 106232611, 106484611, 106565611, 107131711, 107242711, 107585711, 107878711, 107959711, 108000811, 108535811, 108717811, 108858811, 109343911, 109616911, 109737911, 110464021, 110767021, 110797021, 110858021, 111444121, 111646121, 111828121, 111888121, 112020221, 112101221, 112343221, 112535221, 112565221, 112848221, 112929221, 113080321, 113353321, 113494321, 113606321, 114484421, 114636421, 114666421, 115161521, 115595521, 115767521, 116272621, 117030721, 117151721, 117777721, 118151821, 118404821, 118989821, 119252921, 119272921, 119999921, 120141031, 120232031, 120727031, 120868031, 121262131, 121292131, 121404131, 121535131, 121676131, 121808131, 121838131, 122050231, 122191231, 122333231, 122575231, 122989231, 123000331, 123313331, 124101431, 124131431, 124151431, 124535431, 124575431, 124757431, 125090531, 125111531, 125303531, 125555531, 125888531, 126040631, 126070631, 126171631, 126222631, 126676631, 126787631, 127343731, 127707731, 127787731, 128414831, 128636831, 129151931, 129343931, 129585931, 129646931, 129999931, 130434041, 130464041, 130737041, 131323141, 132070241, 132222241, 132262241, 132313241, 132676241, 132757241, 132787241, 132888241, 133161341, 133454341, 133686341, 133707341, 133757341, 133797341, 133989341, 134020441, 134101441, 134272441, 134303441, 134444441, 134494441, 134858441, 134878441, 134909441, 134959441, 135010541, 135070541, 135141541, 135171541, 135363541, 135505541, 135535541, 135666541, 135868541, 136181641, 136353641, 136646641, 136949641, 137030741, 137585741, 137686741, 137797741, 137909741, 137939741, 138060841, 138070841, 138828841, 138868841, 139656941, 139787941, 140858051, 141030151, 141454151, 141595151, 141808151, 141949151, 142020251, 142434251, 142919251, 143050351, 143474351, 143696351, 144030451, 144090451, 144424451, 144646451, 144808451, 144999451, 145070551, 145181551, 145191551, 145434551, 145484551, 146464651, 146575651, 147292751, 147505751, 147636751, 147747751, 147868751, 148191851, 149090951, 149131951, 149434951, 149909951, 150010061, 150535061, 151181161, 151595161, 151737161, 151797161, 152030261, 152141261, 152404261, 152636261, 152848261, 153000361, 153414361, 153666361, 153777361, 153838361, 153939361, 154040461, 154313461, 154323461, 154686461, 154919461, 155101561, 155171561, 155666561, 156010661, 156111661, 156525661, 156636661, 156969661, 157070761, 157202761, 157212761, 157262761, 157686761, 157898761, 158555861, 158959861, 159202961, 159222961, 159292961, 159454961, 159828961, 160040071, 160404071, 160545071, 160656071, 161636171, 161737171, 162040271, 162474271, 162868271, 163070371, 163151371, 163181371, 163484371, 163575371, 163626371, 165030571, 166878671, 167141771, 167191771, 167888771, 167969771, 168484871, 168828871, 169050971, 169151971, 169212971, 169515971, 169959971, 170666081, 170737081, 170969081, 171151181, 171343181, 172181281, 172202281, 173252381, 173383381, 173777381, 174090481, 174232481, 174646481, 175848581, 175919581, 176474681, 176656681, 176888681, 177010781, 177040781, 177565781, 177616781, 177757781, 177939781, 178131881, 178434881, 178616881, 178626881, 179030981, 179111981, 179707981, 181191191, 181797191, 182929291, 182969291, 182999291, 183363391, 184373491, 184434491, 184979491, 185414591, 185848591, 186232691, 186939691, 186979691, 187191791, 187232791, 187535791, 187595791, 187808791, 187949791, 188161891, 188383891, 188696891, 189000991, 189282991, 189505991, 189646991, 300010013, 300232013, 300505013, 300585013, 300717013, 301131113, 301818113, 302555213, 303838313, 303949313, 304202413, 304292413, 304545413, 305080513, 305101513, 305131513, 305333513, 305858513, 305969513, 306070613, 306232613, 306565613, 306939613, 306999613, 307020713, 307727713, 308888813, 309010913, 309282913, 309292913, 309303913, 309565913, 310101023, 310323023, 310353023, 310565023, 310757023, 311171123, 311282123, 312090223, 312141223, 312202223, 312313223, 312858223, 313050323, 313343323, 313434323, 313737323, 313949323, 314020423, 314404423, 314656423, 314686423, 314929423, 314939423, 315171523, 315787523, 315828523, 315858523, 315898523, 316313623, 316383623, 316707623, 316989623, 317101723, 317191723, 317747723, 317909723, 317929723, 318585823, 318676823, 318838823, 318969823, 319232923, 319767923, 320030033, 320404033, 321999133, 322161233, 322202233, 322292233, 322595233, 322707233, 323191333, 323696333, 323717333, 323828333, 324111433, 324373433, 324717433, 324909433, 324919433, 325353533, 325565533, 325797533, 325818533, 326060633, 326191633, 326212633, 326272633, 326444633, 326818633, 326929633, 327303733, 327343733, 327868733, 328868833, 329111933, 329818933, 330414043, 330777043, 330838043, 330969043, 330979043, 331545143, 331919143, 332515243, 332636243, 333070343, 333393343, 333414343, 333696343, 333747343, 333787343, 334121443, 334262443, 334272443, 334565443, 335434543, 335747543, 335777543, 335909543, 336060643, 336282643, 336292643, 336606643, 336646643, 337181743, 337434743, 337484743, 337575743, 337757743, 337797743, 337808743, 337818743, 338060843, 338080843, 338272843, 338333843, 338464843, 338515843, 338777843, 339000943, 339343943, 340818053, 340868053, 341111153, 341222153, 341353153, 341444153, 342373253, 342646253, 342808253, 342898253, 343404353, 343434353, 343676353, 344141453, 344767453, 345616553, 346070653, 346151653, 346202653, 346242653, 346484653, 346767653, 347131753, 347252753, 348141853, 349010953, 349575953, 349979953, 350090063, 350282063, 350909063, 351585163, 351727163, 352040263, 352161263, 353020363, 353434363, 353545363, 353696363, 354282463, 354424463, 354757463, 355101563, 355272563, 355383563, 355898563, 355919563, 356353663, 356383663, 356636663, 356696663, 356747663, 356797663, 356818663, 356959663, 357969763, 358757863, 358959863, 358989863, 359000963, 359060963, 359383963, 359404963, 359434963, 359666963, 360292073, 360898073, 360999073, 361181173, 361242173, 361616173, 362333273, 362545273, 362717273, 362969273, 363313373, 363616373, 363646373, 364212473, 364808473, 364818473, 365464573, 366202673, 367434773, 367686773, 368242873, 368363873, 368848873, 369282973, 369373973, 370262083, 370686083, 370737083, 370818083, 371101183, 372040283, 372363283, 372565283, 372696283, 373464383, 373484383, 373727383, 374686483, 375252583, 375414583, 375535583, 375777583, 375858583, 376424683, 376484683, 376505683, 376535683, 377191783, 377585783, 377828783, 378030883, 378474883, 378787883, 379070983, 379131983, 379434983, 379646983, 379838983, 380030093, 380101093, 380191093, 380626093, 380686093, 380878093, 380969093, 380989093, 381313193, 381919193, 382343293, 382424293, 382838293, 384232493, 384252493, 385434593, 385767593, 385949593, 386545693, 386888693, 386909693, 386959693, 387565793, 387919793, 388484893, 389090993, 389606993, 389848993, 389959993, 700040017, 700272017, 701080117, 701161117, 701585117, 701999117, 702151217, 702444217, 702616217, 702636217, 702696217, 702717217, 702838217, 703212317, 703232317, 703535317, 703878317, 704161417, 704434417, 704909417, 705030517, 705484517, 705757517, 705888517, 705919517, 705979517, 706292617, 706535617, 707131717, 707282717, 707606717, 707737717, 707747717, 708232817, 708252817, 708696817, 709232917, 709484917, 709707917, 709818917, 710161027, 710444027, 710585027, 710717027, 711171127, 711424127, 711666127, 711898127, 711969127, 712010227, 712181227, 712313227, 712515227, 712656227, 713030327, 713464327, 713797327, 714171427, 714282427, 714333427, 714565427, 715040527, 715272527, 715919527, 716777627, 717181727, 717202727, 717909727, 718121827, 718646827, 718989827, 719000927, 719080927, 719272927, 719878927, 720151037, 720282037, 721404137, 721727137, 722212237, 722474237, 722939237, 723393337, 723555337, 724131437, 724505437, 724575437, 724878437, 725090537, 725545537, 725848537, 726444637, 726808637, 726949637, 727161737, 727272737, 727404737, 727535737, 727646737, 727838737, 728020837, 728050837, 728636837, 728656837, 728939837, 729202937, 729222937, 729727937, 730151047, 730353047, 730494047, 730646047, 730848047, 731060147, 731161147, 731191147, 732373247, 733070347, 733202347, 733232347, 733484347, 733595347, 733848347, 734282447, 734636447, 734777447, 734959447, 735252547, 735909547, 735979547, 736232647, 736353647, 736656647, 736878647, 737303747, 737636747, 737666747, 737848747, 737888747, 738202847, 738414847, 739292947, 739434947, 739626947, 740161057, 740444057, 741010157, 741232157, 741252157, 742626257, 742646257, 743252357, 743353357, 744262457, 744313457, 744454457, 744666457, 745131557, 745646557, 746242657, 747262757, 748878857, 748949857, 749141957, 749191957, 749252957, 749282957, 749464957, 750121067, 750282067, 750373067, 750606067, 751565167, 752363267, 752444267, 752555267, 752959267, 753030367, 753919367, 754323467, 754404467, 754434467, 754464467, 754676467, 754707467, 755050567, 755252567, 755323567, 756151667, 757191767, 757262767, 757343767, 757686767, 758464867, 758717867, 758737867, 758747867, 758777867, 758989867, 759787967, 760080077, 760101077, 760646077, 760656077, 761141177, 761222177, 761515177, 761656177, 761797177, 762111277, 762141277, 762151277, 762424277, 763353377, 763676377, 765828577, 767000777, 767111777, 767666777, 767878777, 767888777, 767909777, 767999777, 768121877, 768373877, 768474877, 768939877, 769494977, 769686977, 770050087, 770323087, 770888087, 771000187, 771070187, 771696187, 772070287, 772323287, 772545287, 772979287, 773777387, 774424487, 775080587, 775353587, 775979587, 776030687, 776353687, 776393687, 776404687, 776717687, 776797687, 776828687, 777484787, 777565787, 777595787, 777676787, 778121887, 778424887, 778616887, 778808887, 779242987, 779353987, 779606987, 780111097, 780232097, 780313097, 780343097, 781101197, 781232197, 781343197, 781616197, 781797197, 781818197, 782090297, 782242297, 783171397, 783202397, 783616397, 783888397, 784131497, 784212497, 785050597, 785323597, 785828597, 785999597, 786010697, 786333697, 786393697, 786474697, 786535697, 786868697, 786919697, 786939697, 787080797, 787878797, 788252897, 788323897, 788545897, 788626897, 788888897, 789676997, 789898997, 900222019, 900393019, 900414019, 900595019, 900616019, 900696019, 900777019, 900949019, 900969019, 901797119, 901919119, 902111219, 902878219, 902959219, 902999219, 903808319, 904121419, 904151419, 904272419, 904292419, 905242519, 905404519, 906787619, 906969619, 907505719, 907797719, 907929719, 908161819, 908525819, 908606819, 908626819, 908707819, 908848819, 909363919, 909787919, 909808919, 909979919, 910656029, 910676029, 910707029, 910979029, 911212129, 911393129, 911959129, 912141229, 912555229, 912838229, 913424329, 913707329, 913838329, 913848329, 914393429, 914626429, 915474529, 915595529, 916070629, 916101629, 916424629, 916616629, 916626629, 916767629, 916787629, 916868629, 917252729, 917636729, 917828729, 917888729, 918040829, 918313829, 918919829, 919323929, 919383929, 919797929, 920080039, 920747039, 920797039, 920989039, 921141139, 921454139, 921828139, 921838139, 922080239, 922232239, 923000339, 923020339, 923242339, 923626339, 923797339, 924444439, 925787539, 925878539, 926353639, 926656639, 926777639, 926999639, 927373739, 927393739, 927454739, 927505739, 927747739, 927787739, 927888739, 928232839, 929060939, 929161939, 929525939, 929666939, 930808049, 930919049, 931434149, 932080249, 933040349, 933313349, 933444349, 933777349, 934010449, 934212449, 934232449, 934868449, 934878449, 934919449, 935303549, 935474549, 935606549, 935858549, 936121649, 936222649, 936414649, 936666649, 937383749, 938161849, 938222849, 938606849, 938878849, 939070949, 939595949, 939696949, 939949949, 940494059, 940616059, 940727059, 940868059, 940949059, 941161159, 941656159, 941767159, 941797159, 941878159, 942141259, 942747259, 942979259, 943292359, 943515359, 943656359, 943848359, 943929359, 944020459, 944525459, 944767459, 944989459, 945606559, 945808559, 945939559, 946050659, 946161659, 946262659, 946686659, 947414759, 947494759, 948090859, 948424859, 948606859, 948838859, 948868859, 948969859, 949616959, 949707959, 949727959, 949898959, 949979959, 950080069, 950171069, 950444069, 950878069, 951606169, 951898169, 951939169, 952595269, 953171369, 953212369, 954585469, 954727469, 955878569, 956282669, 956434669, 956969669, 957444769, 957616769, 957898769, 957979769, 958494869, 958505869, 958545869, 959050969, 959414969, 959464969, 959494969, 959515969, 959767969, 959777969, 959818969, 960525079, 960858079, 960888079, 961131179, 961212179, 961292179, 961575179, 961929179, 962171279, 962363279, 962404279, 962777279, 962969279, 964818479, 964929479, 965252579, 965353579, 965909579, 967484779, 967515779, 967595779, 968828879, 968858879, 968878879, 968999879, 969747979, 969777979, 969888979, 970020089, 970202089, 970797089, 971161189, 971282189, 971828189, 971909189, 971959189, 972343289, 972444289, 972454289, 972565289, 973161389, 973212389, 973373389, 973707389, 973898389, 973979389, 974333489, 974353489, 974929489, 974939489, 975141589, 976515689, 977222789, 978444889, 978696889, 978808889, 979737989, 979797989, 979878989, 980737099, 981000199, 981090199, 981303199, 981999199, 982424299, 982505299, 982646299, 982818299, 982838299, 983000399, 983131399, 984696499, 984868499, 985151599, 985181599, 985383599, 985595599, 985707599, 985848599, 986363699, 986525699, 986888699, 987303799, 987676799, 988454899, 989474999, 989494999, 989747999, 989929999

***

Simon wrote:

Here some of many solutions where I picked largest sets / smallest emirps:
Count:2 Diff:18 Emirp set: 13, 79
Count:5 Diff:594 Emirp set: 107, 157, 167, 359, 389
Count:7 Diff:180 Emirp set: 1021, 3023, 3463, 7027, 7577, 7687, 9029
Count:8 Diff:79992 Emirp set: 10009, 10909, 11719, 12829, 14449, 14549, 17579, 18089
Count:11 Diff:82962 Emirp set: 10039, 10639, 10739, 10939, 11149, 12659, 13469, 14479, 14879, 15289, 16699
Count:14 Diff:5940 Emirp set: 10061, 11071, 11971, 13291, 13591, 31873, 32783, 32983, 70667, 73597, 92189, 92489, 92789, 93199
Count:19 Diff:4950 Emirp set: 10151, 10651, 11161, 12071, 13781, 14591, 14891, 30853, 31063, 31963, 32173, 70457, 72577, 74197, 74897, 90059, 92479, 92779, 94399
Count:29 Diff:1980 Emirp set: 10321, 11731, 12241, 12641, 12841, 13151, 13751, 16481, 17491, 30223, 30323, 31033, 32143, 34963, 70327, 71437, 71537, 72047, 72547, 74167, 75577, 76387, 76487, 77797, 93559, 94169, 95279, 95479, 96289
Count:38 Diff:990 Emirp set: 10711, 11621, 13441, 13841, 14251, 14551, 15461, 15661, 17681, 18191, 18691, 31223, 31723, 32233, 32633, 32933, 35363, 36373, 36473, 36973, 38393, 38993, 70717, 72337, 74357, 75167, 75767, 77587, 78697, 90019, 91129, 91229, 92639, 94559, 96179, 97789, 98299, 98999
Count:58 Diff:2299968 Emirp set: 1000033, 1020233, 1077733, 1078733, 1083833, 1099933, 1165643, 1173743, 1175743, 1213153, 1239353, 1247453, 1281853, 1292953, 1307063, 1337363, 1359563, 1364663, 1431373, 1435373, 1494973, 1554583, 1579783, 1588883, 1628293, 1630393, 1644493, 1655593, 7036339, 7071739, 7080839, 7085839, 7149449, 7154549, 7178749, 7216159, 7221259, 7262659, 7265659, 7277759, 7287859, 7296959, 7317169, 7322269, 7336369, 7345469, 7396969, 7412179, 7445479, 7473779, 7505089, 7528289, 7633399, 7660699, 7666699, 7677799, 7683899, 7693999
...
Count:8185 Diff:99999900 Emirp set: 10000800101, 10004540101, 10008480101, 10021212101, 10021412101, 10021512101, 10021712101, 10024242101, 10024842101, 10033133101, 10038683101, 10048284101, 10049094101, 10050905101, 10053435101, 10053735101, 10054245101, 10054545101, 10061016101, 10066166101, 10069796101, 10076267101, 10076367101, 10081518101, 10084148101, 10087478101, 10093539101, 10097579101, 10097679101, 10101310201, 10105550201, 10107470201, 10109290201, 10112921201, 10114241201, 10121512201, 10122622201, 10128082201, 10134343201, 10136263201, 10137373201, 10139093201, 10142824201, 10142924201, 10144744201, 10161316201, 10163036201, 10164046201, 10184048201, 10185958201, 10191619201, 10193339201, 10194949201, 10195259201, 10203430301, 10205250301, 10206460301, 10212021301, 10212721301, 10214541301, 10216261301, 10224742301, 10231413301, 10243234301, 10246764301, 10247574301, 10254645301, 10257675301, 10257775301, 10263636301, 10266666301, 10267576301, 10268686301, 10269996301, 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95307170459, 95311111459, 95316061459, 95317871459, 95320102459, 95333033459, 95347174459, 95350205459, 95352725459, 95353735459, 95355055459, 95358385459, 95361416459, 95370707459, 95374147459, 95375657459, 95383438459, 95387078459, 95393639459, 95400100559, 95400400559, 95410501559, 95411911559, 95414641559, 95415451559, 95418781559, 95423532559, 95431513559, 95433133559, 95438883559, 95444044559, 95445754559, 95447374559, 95450505559, 95452425559, 95453635559, 95455755559, 95456665559, 95466466559, 95469096559, 95470207559, 95478987559, 95481418559, 95492029559, 95496769559, 95497179559, 95503230659, 95508180659, 95526962659, 95534843659, 95544444659, 95547774659, 95550305659, 95560806659, 95564646659, 95565456659, 95569796659, 95574047659, 95585958659, 95605150759, 95611511759, 95612921759, 95618881759, 95621512759, 95624242759, 95627372759, 95645654759, 95647474759, 95648684759, 95649194759, 95649394759, 95650805759, 95653135759, 95654645759, 95657975759, 95658785759, 95666666759, 95671517759, 95673137759, 95677777759, 95680808759, 95681018759, 95685958759, 95688088759, 95695759759, 95695859759, 95700700859, 95708980859, 95714541859, 95719791859, 95719891859, 95727172859, 95734243859, 95740504859, 95742324859, 95744544859, 95745154859, 95751015859, 95756265859, 95762326859, 95777077859, 95784048859, 95787678859, 95796469859, 95800500959, 95806560959, 95808480959, 95809790959, 95816561959, 95819691959, 95819991959, 95828682959, 95841714959, 95842524959, 95850705959, 95850905959, 95858685959, 95874447959, 95877777959, 95879597959, 95881818959, 95892429959, 95893439959, 95898489959, 96001710169, 96005950169, 96006360169, 96009690169, 96009990169, 96029092169, 96030703169, 96034543169, 96041014169, 96041314169, 96045154169, 96049494169, 96062326169, 96070707169, 96072027169, 96072427169, 96080008169, 96083538169, 96086268169, 96091719169, 96098889169, 96101510269, 96103530269, 96108180269, 96114941269, 96116561269, 96116661269, 96125652269, 96133533269, 96138083269, 96142124269, 96142824269, 96146364269, 96157175269, 96160406269, 96167276269, 96168386269, 96170307269, 96172227269, 96175157269, 96176267269, 96179397269, 96198089269, 96201910369, 96210401369, 96225352369, 96233633369, 96236563369, 96239593369, 96251515369, 96252425369, 96254045369, 96261316369, 96262226369, 96263636369, 96271717369, 96273437369, 96278687369, 96280208369, 96288688369, 96290209369, 96293039369, 96293639369, 96295859369, 96305550469, 96308680469, 96311511469, 96313531469, 96317671469, 96323632469, 96327972469, 96328182469, 96334243469, 96339193469, 96340404469, 96351015469, 96355155469, 96360906469, 96362226469, 96365656469, 96366066469, 96369996469, 96370107469, 96371817469, 96372727469, 96374247469, 96375057469, 96375757469, 96376467469, 96379897469, 96385158469, 96389298469, 96390709469, 96393439469, 96394849469, 96398389469, 96398689469, 96410501569, 96411311569, 96411911569, 96412721569, 96419891569, 96422022569, 96434343569, 96439893569, 96442524569, 96451215569, 96456165569, 96465456569, 96469196569, 96470207569, 96473037569, 96473237569, 96475257569, 96478287569, 96483138569, 96510701669, 96512921669, 96514441669, 96516161669, 96516761669, 96521612669, 96530603669, 96535153669, 96547274669, 96552925669, 96554645669, 96557275669, 96559595669, 96566266669, 96568286669, 96569396669, 96570407669, 96579797669, 96587878669, 96589598669, 96595459669, 96600000769, 96600600769, 96603130769, 96607570769, 96622722769, 96625452769, 96633033769, 96636163769, 96639693769, 96642324769, 96645354769, 96651315769, 96651615769, 96655255769, 96664146769, 96667476769, 96669996769, 96670107769, 96677677769, 96677977769, 96678687769, 96689098769, 96691719769, 96693039769, 96699199769, 96705350869, 96706160869, 96712821869, 96714641869, 96720302869, 96720902869, 96726062869, 96732123869, 96744444869, 96746964869, 96747074869, 96753335869, 96761516869, 96761616869, 96765356869, 96768686869, 96775257869, 96788688869, 96803230969, 96804640969, 96825052969, 96826862969, 96840104969, 96856465969, 96863936969, 96867976969, 96870407969, 96874747969, 96879397969, 96881318969, 96885058969, 96887578969, 96888988969, 96892529969, 96899299969, 96899599969, 97002120179, 97004040179, 97009290179, 97016661179, 97024542179, 97029392179, 97032623179, 97032923179, 97038683179, 97040604179, 97051715179, 97054145179, 97055255179, 97067976179, 97068686179, 97074747179, 97075557179, 97077077179, 97079097179, 97082628179, 97087878179, 97099299179, 97099799179, 97099899179, 97100600279, 97102420279, 97102520279, 97103030279, 97107370279, 97108280279, 97116161279, 97117571279, 97117771279, 97118581279, 97118981279, 97119391279, 97121612279, 97126562279, 97127872279, 97128682279, 97128782279, 97130603279, 97134343279, 97135853279, 97151015279, 97155455279, 97165756279, 97173737279, 97175557279, 97176367279, 97179497279, 97187078279, 97188388279, 97188488279, 97191019279, 97207270379, 97208680379, 97209990379, 97214541379, 97218781379, 97224742379, 97224942379, 97233433379, 97233633379, 97237473379, 97245654379, 97250505379, 97257775379, 97260006379, 97260106379, 97270107379, 97275357379, 97295259379, 97305650479, 97309590479, 97313031479, 97315551479, 97318281479, 97319291479, 97328382479, 97329792479, 97331813479, 97332723479, 97334943479, 97343534479, 97348284479, 97355255479, 97361916479, 97362126479, 97374347479, 97379597479, 9737989747

***

Giorgios wrote:

I found a set with 1242 elements and k=9999990.
I am sending a file with 1242 primes p such that:
a) p and reverse(p) are primes.
b) p - reverse(p) = 9999990

(the same that found Emmanuel. See above)

***

Seiji wrote:

701 - 107 = 594
751 - 157 = 594
761 - 167 = 594
953 - 359 = 594
983 - 389 = 594

1201 - 1021 = 180
3203 - 3023 = 180
3643 - 3463 = 180
7207 - 7027 = 180
7757 - 7577 = 180
7867 - 7687 = 180
9209 - 9029 = 180

11701 - 10711 = 990
12611 - 11621 = 990
14431 - 13441 = 990
14831 - 13841 = 990
15241 - 14251 = 990
15541 - 14551 = 990
16451 - 15461 = 990
16651 - 15661 = 990
18671 - 17681 = 990
19181 - 18191 = 990
19681 - 18691 = 990
32213 - 31223 = 990
32713 - 31723 = 990
33223 - 32233 = 990
33623 - 32633 = 990
33923 - 32933 = 990
36353 - 35363 = 990
37363 - 36373 = 990
37463 - 36473 = 990
37963 - 36973 = 990
39383 - 38393 = 990
39983 - 38993 = 990
71707 - 70717 = 990
73327 - 72337 = 990
75347 - 74357 = 990
76157 - 75167 = 990
76757 - 75767 = 990
78577 - 77587 = 990
79687 - 78697 = 990
91009 - 90019 = 990
92119 - 91129 = 990
92219 - 91229 = 990
93629 - 92639 = 990
95549 - 94559 = 990
97169 - 96179 = 990
98779 - 97789 = 990
99289 - 98299 = 990
99989 - 98999 = 990

***

I get 202 pairs of emirps with the same difference 90090:

{{1038211, 1128301}, {1016011, 1106101}, {1692871, 1782961}, {1207111, 1117021}, {1174801, 1084711}, {1377821, 1287731}, {3040313, 3130403}, {1351621, 1261531}, {1201111, 1111021}, {1812271, 1722181}, {1459631, 1369541}, {1695871, 1785961}, {1107101, 1017011}, {1155601, 1065511}, {1447351, 1537441}, {1734281, 1824371}, {3255613, 3165523}, {1682951, 1592861}, {1911281, 1821191}, {1507141, 1417051}, {1120121, 1210211}, {3018013, 3108103}, {1595861, 1685951}, {1249511, 1159421}, {1226311, 1136221}, {1381921, 1291831}, {1142321, 1232411}, {3263713, 3173623}, {1366721, 1276631}, {3414233, 3324143}, {1436431, 1346341}, {1488931, 1398841}, {3751663, 3661573}, {1466551, 1556641}, {1939481, 1849391}, {1624171, 1714261}, {1177621, 1267711}, {1958681, 1868591}, {3214213, 3124123}, {1081711, 1171801}, {1535441, 1445351}, {1188721, 1278811}, {1412051, 1502141}, {1141321, 1231411}, {3769583, 3859673}, {3201113, 3111023}, {1606151, 1516061}, {3468733, 3378643}, {1461731, 1371641}, {7687777, 7777867}, {1736461, 1646371}, {1379821, 1289731}, {9605159, 9515069}, {1753481, 1843571}, {3365543, 3455633}, {3760763, 3670673}, {9258439, 9348529}, {1243511, 1153421}, {3648553, 3558463}, {3083713, 3173803}, {1436251, 1526341}, {7264717, 7174627}, {1810271, 1720181}, {1807171, 1717081}, {3826373, 3736283}, {7084717, 7174807}, {1859491, 1949581}, {1684951, 1594861}, {1379641, 1469731}, {3226133, 3316223}, {1426331, 1336241}, {7600157, 7510067}, {7099817, 7189907}, {9548369, 9638459}, {3237413, 3147323}, {7460737, 7370647}, {3608153, 3518063}, {3515063, 3605153}, {1442531, 1352441}, {1390841, 1480931}, {7229137, 7319227}, {7551647, 7461557}, {7711267, 7621177}, {3365723, 3275633}, {3226313, 3136223}, {3594863, 3684953}, {3474833, 3384743}, {3680953, 3590863}, {7970887, 7880797}, {7177627, 7267717}, {3376643, 3466733}, {3586943, 3496853}, {7900187, 7810097}, {9447539, 9357449}, {7676857, 7586767}, {1826371, 1736281}, {3403133, 3313043}, {3834473, 3744383}, {1925381, 1835291}, {3067513, 3157603}, {7297837, 7387927}, {9768769, 9678679}, {3255433, 3345523}, {1508141, 1418051}, {7131407, 7041317}, {7300127, 7210037}, {7526347, 7436257}, {3542543, 3452453}, {3948583, 3858493}, {7516247, 7426157}, {3984983, 3894893}, {7144507, 7054417}, {3072613, 3162703}, {3364723, 3274633}, {3550643, 3460553}, {3689773, 3779863}, {3874693, 3964783}, {7870697, 7960787}, {7167707, 7077617}, {3706163, 3616073}, {7684777, 7774867}, {7191827, 7281917}, {9616259, 9526169}, {7243517, 7153427}, {7875697, 7965787}, {7111207, 7021117}, {3539443, 3449353}, {1310041, 1400131}, {7652657, 7562567}, {3717083, 3807173}, {7279817, 7189727}, {7786967, 7696877}, {7122307, 7032217}, {9700169, 9610079}, {7515247, 7425157}, {3875693, 3965783}, {9859679, 9769589}, {7788787, 7878877}, {9904189, 9814099}, {3758483, 3848573}, {9643559, 9553469}, {3184903, 3094813}, {9649559, 9559469}, {9230239, 9320329}, {3447533, 3357443}, {7838477, 7748387}, {3662753, 3572663}, {7254617, 7164527}, {9034219, 9124309}, {7268717, 7178627}, {7598867, 7688957}, {9080719, 9170809}, {9507149, 9417059}, {9830479, 9740389}, {9329149, 9419239}, {9331429, 9241339}, {7183727, 7273817}, {3637453, 3547363}, {3471833, 3381743}, {3488753, 3578843}, {9259619, 9169529}, {9109109, 9019019}, {9744569, 9654479}, {1619071, 1709161}, {3624353, 3534263}, {9056419, 9146509}, {9179809, 9089719}, {7649557, 7559467}, {9422159, 9512249}, {9826379, 9736289}, {1240511, 1150421}, {3187903, 3097813}, {9889799, 9979889}, {7370827, 7280737}, {3523343, 3433253}, {1974881, 1884791}, {9577849, 9487759}, {7377647, 7467737}, {7542547, 7452457}, {9585769, 9675859}, {9539269, 9629359}, {7371827, 7281737}, {7729367, 7639277}, {9534269, 9624359}, {9893899, 9983989}, {9136229, 9226319}, {7570847, 7480757}, {3577663, 3667753}, {7265537, 7355627}, {3231233, 3321323}, {9883799, 9973889}, {9397849, 9487939}, {3575843, 3485753}, {3741383, 3831473}, {3788963, 3698873}, {9593869, 9683959}, {9787969, 9697879}, {7332427, 7242337}, {9427339, 9337249}, {7565567, 7655657}, {7052417, 7142507}, {9247339, 9337429}}

***

Hakan wrote:

That's 202 emirps set with the same k=90090 value:

1016011,1017011,1038211,1065511,1081711,1084711,1111021,1117021,1120121,1136221,1141321,1142321,1150421,1153421,1159421,
1177621,1188721,1261531,1276631,1287731,1289731,1291831,1310041,1336241,1346341,1352441,1369541,1371641,1379641,1390841,
1398841,1412051,1417051,1418051,1436251,1445351,1447351,1466551,1516061,1592861,1594861,1595861,1619071,1624171,1646371,
1692871,1695871,1717081,1720181,1722181,1734281,1736281,1753481,1821191,1835291,1849391,1859491,1868591,1884791,3018013,
3040313,3067513,3072613,3083713,3094813,3097813,3111023,3124123,3136223,3147323,3165523,3173623,3226133,3231233,3255433,
3274633,3275633,3313043,3324143,3357443,3365543,3376643,3378643,3381743,3384743,3433253,3449353,3452453,3460553,3485753,
3488753,3496853,3515063,3518063,3534263,3547363,3558463,3572663,3577663,3590863,3594863,3616073,3661573,3670673,3689773,
3698873,3717083,3736283,3741383,3744383,3758483,3769583,3858493,3874693,3875693,3894893,7021117,7032217,7041317,7052417,
7054417,7077617,7084717,7099817,7153427,7164527,7174627,7177627,7178627,7183727,7189727,7191827,7210037,7229137,7242337,
7265537,7280737,7281737,7297837,7370647,7377647,7425157,7426157,7436257,7452457,7461557,7480757,7510067,7559467,7562567,
7565567,7586767,7598867,7621177,7639277,7684777,7687777,7696877,7748387,7788787,7810097,7870697,7875697,7880797,9019019,
9034219,9056419,9080719,9089719,9136229,9169529,9230239,9241339,9247339,9258439,9329149,9337249,9357449,9397849,9417059,
9422159,9487759,9515069,9526169,9534269,9539269,9548369,9553469,9559469,9585769,9593869,9610079,9654479,9678679,9697879,
9736289,9740389,9769589,9814099,9883799,9889799,9893899

***

Metin wrote:

9989 is the largest non palindromic n such that abs(n-reversal(n))<=90 .Since there isn't another emirp with that property up 9989, 113,709,739,769 are the only examples.

For n>=98 ;
n     Min abs(n-reversal(n))
98      9
998     99
9989    90
99989   990
999899  900
9999899 9900
...
.... Min abs(n-r(n)) :  9, 90, 900, ... when n has even number of digits.
.... Min abs(n-r(n)) :  99, 990, 9900, ... when n has odd number of digits.

***

Oscar wrote:

Dear Carlos,
for k=198, we also have the difference 311-113.
However, we still have larger sets both for k=90 and for k=180.
Here is a list of the first few sets with common difference k and champion size.

k=18, size=2
31-13
97-79

k=90, size=5
1321-1231
3433-3343
7547-7457
9439-9349
9769-9679

k=180, size=7
1201-1021
3203-3023
3643-3463
7207-7027
7757-7577
7867-7687
9209-9029

k=900, size=34
106501-105601
112111-111211
121021-120121
138731-137831
145441-144541
151051-150151
168761-167861
172171-171271
181081-180181
186581-185681
306503-305603
334333-333433
336533-335633
343243-342343
346543-345643
354353-353453
376573-375673
384383-383483
715417-714517
718717-717817
747647-746747
751057-750157
763267-762367
768767-767867
775477-774577
778777-777877
784387-783487
793297-792397
909809-908909
936539-935639
937639-936739
939839-938939
957659-956759
996599-995699

k=990, size=38
11701-10711
12611-11621
14431-13441
14831-13841
15241-14251
15541-14551
16451-15461
16651-15661
18671-17681
19181-18191
19681-18691
32213-31223
32713-31723
33223-32233
33623-32633
33923-32933
36353-35363
37363-36373
37463-36473
37963-36973
39383-38393
39983-38993
71707-70717
73327-72337
75347-74357
76157-75167
76757-75767
78577-77587
79687-78697
91009-90019
92119-91129
92219-91229
93629-92639
95549-94559
97169-96179
98779-97789
99289-98299
99989-98999

Then the pattern seems to repeat:
k=900, size=34, near-palprimes with length 6

k=990, size=38, near-palprimes with length 5

k=9000, size=131, near-palprimes with length 8
k=9900, size=162, near-palprimes with length 7
k=90000, size=923, near-palprimes with length 10
k=99000, size=974, near-palprimes with length 9

I only checked sets with common difference k<10^5.
Attached file "P1042_champions_OV.txtcontains the nine champions above.

***

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