"Squares containing at most three distinct digits"
[ Patrick De Geest ]

N.J.A. Sloane (2001), Part of the
On-Line Encyclopedia of Integer Sequences

012	013	014	015	016	017	018	019	024	025
029	034	036	039	045	046	047	048	049	056
059	067	069	079	089	123	124	125	126	127
128	129	134	136	138	145	146	147	148	149
156	158	167	168	169	178	189	234	235	236
239	245	246	247	248	249	256	257	258	259
267	269	279	289	345	346	347	348	349	356
367	368	369	389	456	457	458	459	467	468
469	478	479	489	567	568	569	589	678	679
689	789

LEGEND
References LINK 1 : http://arxiv.org/abs/2112.00444 ( Sascha Kurz and three of his students ) LINK 2 : http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm ( Hisanori Mishima's website ) LINK 3 : http://djm.cc/rpa-output/arithmetic/digits/squares/three.digits.s LINK 4 : https://erich-friedman.github.io/mathmagic/0999.html LINK 5 : http://www.worldofnumbers.com/em_crump.htm LINK 6 : http://blue.kakiko.com/mmrmmr/htm/eqtn06.html LINK 7 : http://www.immortaltheory.com/NumberTheory/TriDigitalSquares.htm LINK 8 : http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series002 LINK 9 : http://mathforum.org/rec_puzzles_archive/arithmetic/part2 LINK 10a : https://bbs.emath.ac.cn/forum.php?mod=redirect&goto=findpost&ptid=19296&pid=99359 LINK 10b : https://github.com/emathgroup/selectedTopics/tree/master/data
TRIPLES	ROOT	SQUARE	SOURCE	PATTERNS & SPORADIC RECORDS (mostly L⩾17)
0 1 2	A058411	A058412	LINK 2 LINK 1 LINK 10	Four infinite patterns 1(0n)1 2 = 1(0n)2(0n)1  [n⩾0] 1(0n)11 2 = 1(0n)22(0n-1)121  [n⩾1] 11(0n)1 2 = 121(0n-1)22(0n)1  [n⩾1] 101(0n)10401(0n)101 2 = 10201(0n-2)2101002(0n-4)110221001(0n-4)2101002(0n-2)10201  [n⩾4] [17] 10959977245460011 2 = [33] 120121101221001210210111000120121 [18] 110000500908955011 2 = [35] 12100110200221012201210212022010121 [20] 10099510939154979751 2 = [39] 102000121210111101102120011101220022001 [24] 471287714788971663493899 2 = [48] 222112110111011100020110111110102200012010222201 [29] 10000009999995510010001000001 2 = [57] 100000200000010200110220220200010211120011021020002000001
0 1 3			LINK 2 LINK 10	( No rootsolutions less than 1024 info from Hisanori Mishima's website )
0 1 4	A058413	A058414	LINK 2 LINK 10	Six infinite patterns 1(0n)2 2 = 1(0n)4(0n)4  [n⩾0] 2(0n)1 2 = 4(0n)4(0n)1  [n⩾0] 102(0n)201 2 = 10404(0n-2)41004(0n-2)40401  [n⩾2] 1(0n)202(0n)1 2 = 1(0n)404(0n-2)41004(0n-2)404(0n)1  [n⩾2] 201(0n)102 2 = 40401(0n-2)41004(0n-2)10404  [n⩾2] 2(0n)101(0n)2 2 = 4(0n)404(0n-2)11001(0n-2)404(0n)4  [n⩾2] [17] 10677612092787462 2 = [33] 114011400004041044011001104401444 [18] 105423154192999799 2 = [35] 11114041440001011101141014414040401 [19] 3743127183788194652 2 = [38] 14011001114014141144414101441441401104 [22] 3180252254777039538502 2 = [44] 10114004404014444004140001011411401140404004
0 1 5	A058415	A058416	LINK 2 LINK 10	[17] 23452400954944999 2 = [33] 550015110551505101015151115110001
0 1 6	A058417	A058418	LINK 2 LINK 1 LINK 10	Three infinite patterns 1(0n)3(0n)1 2 = 1(0n)6(0n-1)11(0n)6(0n)1  [n⩾1] 1(0n)8(0n)1 2 = 1(0n-1)16(0n-1)66(0n-1)16(0n)1  [n⩾1] 4(0n)127(0n+2)4 2 = 16(0n-1)1016(0n-2)16161(0n-1)1016(0n+1)16 [n⩾2] [17] 12649351807945204 2 = [33] 160006101161166601100660666601616 [26] 77470059130002034719700749 2 = [52] 6001610061606011616611060006010661000616100111161001
0 1 7	A058419	A058420	LINK 2 LINK 10	[16] 8427200114569499 2 = [32] 71017701771000177071770101111001
0 1 8	A058421	A058422	LINK 2 LINK 10	Four infinite patterns 9(0n)1 2 = 81(0n-1)18(0n)1  [n⩾1] 1(0n+1)9 2 = 1(0n)18(0n)81  [n⩾0] 1(0n)4(0n)1 2 = 1(0n)8(0n-1)18(0n)8(0n)1  [n⩾1] 1(0n+1)9(0n)1 2 = 1(0n)18(0n-1)101(0n-1)18(0n)1  [n⩾1] [18] 283160940117244651 2 = [35] 80180118008081811188010188188111801 [18] 331680389653656009 2 = [36] 110011880880801080101881180101808081 [28] 2981276371121751737986262751 2 = [55] 8888008801008880800188080010010188818118011188010088001
0 1 9	A058473	A058474	LINK 2 LINK 10	[20] 43694278824566964251 2 = [40] 1909190001999001011109190090109911991001 [27] 100990098979999970099500001 2 = [53] 10199000091990191001091091099001091999900190199000001
0 2 3	-	-	-	combination impossible
0 2 4	A058423	A058424	LINK 2 LINK 1 LINK 10	One infinite pattern 2(0n)6(0n)2 2 = 4(0n-1)24(0n-1)44(0n-1)24(0n)4  [n⩾1] [16] 1562062816343832 2 = [31] 2440040242204024220420044444224 [24] 205524700326856587391168 2 = [47] 42240402444444204240022400420200244000244404224 [27] 634050802727999251005000002 2 = [54] 402020420440020222440222024020044204422022004020000004
0 2 5	A058425	A058426	LINK 2 LINK 1 LINK 10	Four infinite patterns 5(0n)5 2 = 25(0n-1)5(0n)25  [n⩾1] 5(0n+1)505 2 = 25(0n)505(0n-1)255025  [n⩾1] 505(0n+2)5 2 = 255025(0n-1)505(0n+2)25  [n⩾1] 15(0n+2)85(0n)15 2 = 225(0n)255(0n)52225(0n-2)255(0n)225  [n⩾2] Farideh Firoozbakht's intricate patterns [17] 14899671139156245 2 = [33] 222000200055005555555250522500025 [17] 50000050049995505 2 = [34] 2500005005002055502050050520205025 [20] 23500043724538482665 2 = [39] 552252055055220520520522052500505502225 [20] 23558545158870178045 2 = [39] 555005050002525502502022522050000022025 [21] 150167406766664999985 2 = [41] 22550250055025025225250200022000050000225 [22] 5000000500000500254955 2 = [44] 25000005000005252550050255205255020002052025 [22] 5000005004997549999955 2 = [44] 25000050050000550000025505552050220500002025 [23] 44949994999999949999995 2 = [46] 2020502050500020505000050500052500000500000025 [24] 447241797269721007814765 2 = [48] 200025225225050225520202505522022522200552005225 [25] 5000000500499975499999955 2 = [50] 25000005005000005500225025500555205002205000002025 [27] 500000000000500000500254955 2 = [54] 250000000000500000500255205000500255205255020002052025 [29] 50000005004999999999955050005 2 = [58] 2500000500500025050020505000050050550050002020502050500025 [29] 50050000004999999999955050005 2 = [58] 2505002500500500000020500505500050500050002020502050500025 [30] 141422082876067219949805050005 2 = [59] 20000205525005225202000505202222520222202205205000550500025
0 2 6	-	-	-	combination impossible
0 2 7	-	-	-	combination impossible
0 2 8	-	-	-	combination impossible
0 2 9	A058427	A058428	LINK 2 LINK 10	[17] 96385117673990673 2 = [34] 9290090909029029202290909290992929 [18] 151400161921673747 2 = [35] 22922009029909029220029029909020009 [21] 149067065510873088673 2 = [41] 22220990020022929092929022220290920900929
0 3 4	A058429	A058430	LINK 2 LINK 10	[19] 5773251280200207952 2 = [38] 33330430344333340030340440344044034304 [23] 20832739723817975138362 2 = [45] 434003044400343443044430000434430333044043044
0 3 5	-	-	-	combination impossible
0 3 6	A058431	A058432	LINK 2 LINK 10	[17] 25107103902348156 2 = [33] 630366666363306003336330636600336
0 3 7	-	-	-	combination impossible
0 3 8	-	-	-	combination impossible
0 3 9	A058433	A058434	LINK 2 LINK 10	[9] 969071253 2 = [18] 939099093390990009
0 4 5	A058435	A058436	LINK 2 LINK 1 LINK 10	[20] 21082112192576920612 2 = [39] 444455454500400455000455440400550454544 [20] 63639964217112858462 2 = [40] 4050045045555405040055544045540445005444 [22] 6674983479713230005962 2 = [44] 44555404454444540454555540045000554555545444 [26] 21214250022106461574572502 2 = [51] 450044404000444005405445044455050544404404054540004 [28] 2108436491907081488939581538 2 = [55] 4445504440405440505004450045555054500055550554550445444
0 4 6	A058437	A058438	LINK 2 LINK 10	One infinite pattern 8(0n)254(0n+2)8 2 = 64(0n-1)4064(0n-2)64644(0n-1)4064(0n+1)64  [n⩾2] [18] 635613598504629262 2 = [36] 404004646604004046006460044066664644 [19] 2457776365832743262 2 = [37] 6040664664446006640606666600406400644 [19] 2542962918459579238 2 = [37] 6466660404660460644444606044000660644
0 4 7	A058439	A058440	LINK 2 LINK 10	[16] 2010988315424552 2 = [31] 4044074004774077447400004400704
0 4 8	A058441	A058442	LINK 2 LINK 1 LINK 10	Five infinite patterns 2(0n)2 2 = 4(0n)8(0n)4  [n⩾0] 2(0n)2(0n-1)2 2 = 4(0n)8(0n-1)84(0n-1)8(0n-1)4  [n⩾1] 2(0n-1)2(0n)2 2 = 4(0n-1)8(0n-1)48(0n-1)8(0n)4  [n⩾1] 2(0n)22 2 = 4(0n)88(0n-1)484  [n⩾1] 22(0n)2 2 = 484(0n-1)88(0n)4  [n⩾1] [17] 21919954490920022 2 = [33] 480484404884004840840444000480484 [18] 220001001817910022 2 = [35] 48400440800884048804840848088040484 [20] 20199021878309959502 2 = [39] 408000484840444404408480044404880088004 [24] 942575429577943326987798 2 = [48] 888448440444044400080440444440408800048040888804 [29] 20000019999991020020002000002 2 = [57] 400000800000040800440880880800040844480044084080008000004
0 4 9	A058443	A058444	LINK 2 LINK 10	Two infinite patterns (9n)7 2 = (9n)4(0n)9  [n⩾1] 2(0n)1(0n)2 2 = 4(0n)4(0n)9(0n)4(0n)4  [n⩾0] [17] 99704560597822753 2 = [34] 9940999404004909449099404004499009 [22] 3015775265159011230138 2 = [43] 9094900449944904494440090444449999999499044
0 5 6	A058445	A058446	LINK 2 LINK10	[16] 2236081408416666 2 = [31] 5000060065066660656065066555556
0 5 7	-	-	-	combination impossible
0 5 8	-	-	-	combination impossible
0 5 9	A058447	A058448	LINK 2 LINK 10	[11] 70778174997 2 = [22] 5009550055905955950009 [12] 771395165003 2 = [24] 595050500590005595990009
0 6 7	A058449	A058450	LINK 2 LINK 10	[20] 26012881552428213576 2 = [39] 676670006660660066767076066770670707776
0 6 8	-	-	-	combination impossible
0 6 9	A058451	A058452	LINK 2 LINK 10	[17] 30000101109940614 2 = [33] 900006066606660060090966606696996 [29] 24691314243454114014126412353 2 = [57] 609660999069000006699096996606066090996096009966990996609
0 7 8	-	-	-	combination impossible
0 7 9	A058453	A058454	LINK 2 LINK 10	[16] 8819172285373497 2 = [32] 77777799799099990007000790009009
0 8 9	A058455	A058456	LINK 2 LINK 10	[15] 301345331969667 2 = [29] 90809009099908808089808090889 [27] 299831600904572582192518303 2 = [53] 89898988900998890088080098089989880890999988989999809
1 2 3	A030175	A030174	LINK 2 LINK 10	[20] 56843832676142723489 2 = [40] 3231221313313311221231322223122312333121 [21] 557963558954625926861 2 = [42] 311323333121312322332133323111223321313321
1 2 4	A053880	A053881	LINK 2 LINK 10	One infinite pattern (3n)8 2 = (1n)4(2n-1)44  [n⩾1] [16] 4705573731461671 2 = [32] 22142424142222114221142142112241 [21] 379766258564954821662 2 = [42] 144222411144424121442444112111142224442244 [28] 1114110597927523626433041668 2 = [55] 1241242424414424212214142144114212224412421422224222224
1 2 5	A031153	A031153	LINK 2 LINK 10	Three infinite patterns (3n)5 2 = (1n)(2n+1)5  [n⩾0] 123(3n)5 2 = 152(1n)5(2n+2)5  [n⩾0] (3n)504485 2 = (1n)2251(2n-4)51515115225  [n⩾4] [18] 159722338802442489 2 = [35] 25511225512522225551152512152515121 [21] 124588166420914599285 2 = [41] 15522211212125512112255222511252122511225 [21] 150051695267169681235 2 = [41] 22515511252551552115222525221211511125225 [23] 34816980372445012123335 2 = [46] 1212222122255221215111111512151155125251522225 [25] 1102340925268369741032335 2 = [49] 1215155515521525522215211555521512552151515552225 [28] 4638162516046117503822620335 2 = [56] 21512551525255251211125255525551555121211151225555512225
1 2 6	A053882	A053883	LINK 2 LINK 10	[20] 47130268582155593596 2 = [40] 2221262216626122626662262611111116211216 [26] 51595698572871617009432954 2 = [52] 2662116111222626216162621266666216222222216621166116
1 2 7	A053884	A053885	LINK 2 LINK 10	[18] 130834904430015239 2 = [35] 17117772217211221211117217772227121
1 2 8	A053886	A053887	LINK 2 LINK 10	One infinite pattern (3n)59 2 = (1n)28(2n-1)881  [n⩾1] [17] 28616100692061141 2 = [33] 818881218818182112888822882221881 [17] 33350084135098989 2 = [34] 1112228111818181281181888828822121 [19] 1490909832561385391 2 = [37] 2222812128828218222281282181228222881 [20] 34377642169166984891 2 = [40] 1181822281111288118221882821111822281881
1 2 9	A053888	A053889	LINK 2 LINK 10	[17] 99961098929489127 2 = [34] 9992221299191112291912929211222129 [18] 459556524411439511 2 = [36] 211192199129121999192121999211919121 [27] 145303131149776986249167839 2 = [53] 21112999921929291129929211912291229929191119991929921
1 3 4	A053890	A053891	LINK 2 LINK 10	[20] 21079405433537116521 2 = [39] 444341333431434111311131411343131143441 [30] 177324875114669443080086908188 2 = [59] 31444111334433114334141133143444444313434111431113141443344
1 3 5	-	-	-	impossible since all 3 digits odd
1 3 6	A053892	A053893	LINK 2 LINK 10	[17] 18266544874814631 2 = [33] 333666661663616663311166611666161 [29] 11537606482136410218512760694 2 = [57] 133116363336636111166666336113333136136363666113311361636
1 3 7	-	-	-	impossible since all 3 digits odd
1 3 8	A053894	A053895	LINK 2 LINK 10	[24] 286074095527510693610891 2 = [47] 81838388131883313833383381811133318888113813881
1 3 9	-	-	-	impossible since all 3 digits odd
1 4 5	A053896	A053897	LINK 2 LINK 10	[14] 73560479506012 2 = [28] 5411144145154411455544144144
1 4 6	A027677	A027676	LINK 2 LINK 9 LINK 10	[19] 1290017214657004546 2 = [37] 1664144414111416144464164661464666116 [21] 375720828696801774892 2 = [42] 141166141116611464114116414644641441611664 [21] 802565644925350914229 2 = [42] 644111614414444441664646611416446114664441 [22] 4051067284576580130696 2 = [44] 16411146144166666464466464146441416441444416 [23] 10789398111648380852704 2 = [45] 116411111611641646616166611414441166144111616 [30] 105567345643273687982611367608 2 = [59] 11144464466166416111166414141141166144146161144464111641664
1 4 7	A053898	A053899	LINK 2 LINK 10	[19] 1071033028175028538 2 = [37] 1147111747441771474111741117114417444 [22] 2177492084289725902412 2 = [43] 4741471777144414774147714111744447747417744
1 4 8	A053900	A053901	LINK 2 LINK 10	[18] 284800189808379191 2 = [35] 81111148114888814414411114441814481 [19] 2117642891144336478 2 = [37] 4484411414414144114118188814881444484 [19] 6435147099182306059 2 = [38] 41411118188114448414441181181148111481 [25] 1346900557360669225841779 2 = [49] 1814141111418481411488184148411114184811141884841 [28] 2209483759119790145920022988 2 = [55] 4881818481814118844811411488844844184118148818448448144
1 4 9	A027675	A006716 * (* seq by Neil Sloane)	LINK 2 LINK 9 LINK 4 LINK 10	[18] 648070211589107021 2 = [36] 419994999149149944149149944191494441
1 5 6	A053902	A053903	LINK 2 LINK 10	One infinite pattern (3n)4 2 = (1n+1)(5n)6  [n⩾0] [17] 12472031176057954 2 = [33] 155551561656561551165551166666116 [19] 2482270463831785216 2 = [37] 6161666655611666116165616661556166656 [20] 12516036335682176169 2 = [39] 156651165556116515661615565555551516561 [21] 258100003219026842869 2 = [41] 66615611661661666651111615111161616151161 [23] 74240565428619726479296 2 = [46] 5511661555161166511651661611666516615516655616
1 5 7	-	-	-	impossible since all 3 digits odd
1 5 8	A053904	A053905	LINK 2 LINK 10	[19] 3399291958357679641 2 = [38] 11555185818155188818511188881585888881 [22] 2412237158970509643109 2 = [43] 5818888111118115811551585855811558551185881
1 5 9	-	-	-	impossible since all 3 digits odd
1 6 7	A053906	A053907	LINK 2 LINK 10	[10] 1292931424 2 = [19] 1671671667166667776
1 6 8	A053908	A053909	LINK 2 LINK 10	[17] 12723468913060546 2 = [33] 161886661181618111866616661818116 [18] 126931550889393381 2 = [35] 16111618611186661611161686166611161 [21] 431495267861269619604 2 = [42] 186188166186668818681818186188818861116816 [25] 1080794204132598414568541 2 = [49] 1168116111686616811868118886618818666111186868681
1 6 9	A053910	A053911	LINK 2 LINK 10	[19] 3019927482025216937 2 = [37] 9119961996691166966116961161911661969 [19] 4082512947923373236 2 = [38] 16666911969961991191619191116961111696 [21] 411900436901564744737 2 = [42] 169661969919699919691616999616691969199169 [23] 34149670012924966713187 2 = [46] 1166199961991666696199696161116991961919696969 [30] 248216864092061020657513399437 2 = [59] 61611611619696691696969619616966119999999161919199911916969
1 7 8	A053912	A053913	LINK 2 LINK 10	[9] 105769141 2 = [17] 11187111187877881 [9] 279067891 2 = [17] 77878887787187881
1 7 9	-	-	-	impossible since all 3 digits odd
1 8 9	A053914	A053915	LINK 2 LINK 10	[16] 2969848344609859 2 = [31] 8819999189981919818818919999881
2 3 4	A053916	A053917	LINK 2 LINK 10	[15] 205483392086668 2 = [29] 42223424423443333243223342224
2 3 5	A053918	A053919	LINK 2 LINK 10	[18] 159789024443333515 2 = [35] 25532532332552235533223325522255225 [27] 576387476638096486959455635 2 = [54] 332222523225232223533222222253253255253352335533253225
2 3 6	A058457	A058458	LINK 2 LINK 10	[16] 2514602599284156 2 = [31] 6323226232326633633323632632336 [27] 251462176552105392823457806 2 = [53] 63233226236322223226263332266636366232262662262333636
2 3 7	-	-	-	combination impossible
2 3 8	-	-	-	combination impossible
2 3 9	A053920	A053921	LINK 2 LINK 10	[19] 1814641285211195673 2 = [37] 3292922993992939999922923222293922929 [20] 14940646884386874573 2 = [39] 223222929323939222223332232999233932329
2 4 5	A031154	A031152	LINK 2 LINK 10	Three infinite patterns (6n)5 2 = (4n)(2n+1)5  [n⩾0] (6n)515 2 = (4n)24(2n-1)45225  [n⩾1] 2(3n)5 2 = 5(4n-1)5(2n+1)5  [n⩾1] [22] 1566985170463509665838 2 = [43] 2455442524452554445254444455245254424242244 [22] 5022405325580985229335 2 = [44] 25224555254424242244454444254455442544542225 [22] 5042276720337304556485 2 = [44] 25424554524455524225542255422225542555555225 [24] 156602211177063382566485 2 = [47] 24524252545545555425452252442555424225445255225 [24] 473545618135383472428338 2 = [48] 224245452455222424254455242452522555442545442244 [27] 212706944938912242495946332 2 = [53] 45244244425245444452555552444225545225555452224254224
2 4 6	A053922	A053923	LINK 2 LINK 10	One infinite pattern (6n)8 2 = (4n)6(2n)4  [n⩾0] [17] 14988743331128338 2 = [33] 224662426646444226244244226642244 [18] 211335426908131668 2 = [35] 44662662666442262666444262424462224 [19] 4964540927663570432 2 = [38] 24646666622446664464662466646224666624 [22] 8015264452445860907338 2 = [44] 64244464242642246466444266646244264622246244 [25] 1562826497869300353470568 2 = [49] 2442426662442422262266222264624646466242442242624 [26] 47144739098275455895604568 2 = [52] 2222626424644462446266642442624422642664422222466624 [27] 163176169897520398456349838 2 = [53] 26626462422424442244464226222466224666246222642626244 [30] 162062538622046218465618335432 2 = [59] 26264266424622222222622644266266266222662426642466466626624
2 4 7	A058459	A058460	LINK 2 LINK 10	[11] 88002411582 2 = [22] 7744424444247727742724 [17] 47146022358675418 2 = [34] 2222747424244722424422447477474724
2 4 8	A027679	A027678	LINK 2 LINK 9 LINK 8 LINK 10	[18] 669644852476481662 2 = [36] 448424228448248888288442822222282244 [20] 20597146608802018338 2 = [39] 424242448424484484244824228422488282244 [23] 53710727465081333454522 2 = [46] 2884842244828242284244242842824284482242248484 [25] 2069416058768323727702022 2 = [49] 4282482824288222284288288882288482848444822888484 [27] 149140498954591218312271662 2 = [53] 22242888428424424282282224442842448448442222888242244 [29] 20598117118436403877526792022 2 = [58] 424282428824822822284282828848288484482824448422442848484
2 4 9	A053924	A053925	LINK 2 LINK 10	[21] 222468490448488507807 2 = [41] 49492229242429222429494944494949499949249 [21] 547259530974381470838 2 = [42] 299492994242299992492444422992424244422244
2 5 6	A030486	A030484	LINK 2 LINK 10	[18] 237112320688458875 2 = [35] 56222252622266562625655622566265625 [18] 257395128676171075 2 = [35] 66252252266222665256252522666655625 [18] 745355990494250516 2 = [36] 555555552565665265566665252566266256 [19] 7453363177489241484 2 = [38] 55552622655552522252252226565666522256 [20] 80781591501545428925 2 = [40] 6525665525522556666225656625562226655625 [21] 228618071522411733125 2 = [41] 52266222626626566662522622656666222265625 [24] 237750344316525128700475 2 = [47] 56525226222626252566552565525652262562265225625 [24] 516291221749568609228465 2 = [48] 266556625655662226525525225552225265562566256225 [26] 81401637345465395512991484 2 = [52] 6626226562522666562566262626266252566552622656522256 [29] 14920674457351323857264196585 2 = [57] 222626526262256222655556652225555252265566656525525662225 [29] 23508012597117321085533117075 2 = [57] 552626656266226655522226225556662256522626266565656555625
2 5 7	A030487	A030485	LINK 2 LINK 10	One infinite pattern 1(6n)5 2 = 2(7n)(2n+1)5  [n⩾0] [18] 870185357137045415 2 = [36] 757222555775727275772757755772522225 [26] 52174924557278712520943915 2 = [52] 2722222752557725255755757775775772222257522575527225
2 5 8	A053926	A053927	LINK 2 LINK 10	[18] 159013392166264585 2 = [35] 25285258888222255288288552225222225
2 5 9	A053928	A053929	LINK 2 LINK 10	[17] 77170285247817565 2 = [34] 5955252925229529299525595522529225 [19] 9794365502654705173 2 = [38] 95929595599592555525252922555552959929 [22] 1598601020441309256565 2 = [43] 2555525222555995255555255252529952995599225 [26] 30484348812550551609088485 2 = [51] 929295522525252225925225599299529559999252559595225 [26] 76975943837016723668817565 2 = [52] 5925295929599552922952299222959292529595925252529225
2 6 7	A058461	A058462	LINK 2 LINK 10	[18] 150573163864701424 2 = [35] 22672277676226226672276776667627776
2 6 8	-	-	-	combination impossible
2 6 9	A053930	A053931	LINK 2 LINK 7 LINK 5 LINK 1 LINK 10	[20] 78883604126137785577 2 = [40] 6222622999929222269696699966269229222929 [23] 47567102808870567435673 2 = [46] 2262629269629662226292666922999622262992962929 [26] 26395073915340646948470264 2 = [51] 696699926996296229992699669262629222696929692229696
2 7 8	-	-	-	combination impossible
2 7 9	A053932	A053933	LINK 2 LINK 10	[11] 14907304327 2 = [21] 222227722297792922929 [27] 850281851974525726895170673 2 = [54] 722979227797229279772792797979272222799797729799272929
2 8 9	A053934	A053935	LINK 2 LINK 10	[13] 5405829167667 2 = [26] 29222988989999289998222889 [26] 17320185602062360469701767 2 = [51] 299988829289888292222928892882898892999989922922289
3 4 5	A053936	A053937	LINK 2 LINK 10	[20] 21343231796858797962 2 = [39] 455533543534444433554534333443535353444
3 4 6	A053938	A053939	LINK 2 LINK 10	[17] 66817996351009092 2 = [34] 4464644636363464333646646666664464 [19] 6666680833328031344 2 = [38] 44444633333463334436443663666646446336
3 4 7	A058463	A058464	LINK 2 LINK 10	[17] 21150351639576462 2 = [33] 447337374477734734334374744437444 [20] 20913496712251033188 2 = [39] 437374344733334774447744773373477443344 [22] 1864878916830083039312 2 = [43] 3477773374437343773773333743737447337433344 [27] 271006150065722262703289312 2 = [53] 73444333373444774773333473377377773344743344373433344
3 4 8	A053940	A053941	LINK 2 LINK 10	[19] 6952948212333013522 2 = [38] 48343488843384848488834343333834844484 [25] 5906300402396058566810062 2 = [50] 34884384443343843348888488484833488344838384443844
3 4 9	A053942	A053943	LINK 2 LINK 10	[21] 185605616817891584607 2 = [41] 34449444994349999433343349994949439344449
3 5 6	A053944	A053945	LINK 2 LINK 10	[17] 23527926717739784 2 = [33] 553563335635333565566365536366656
3 5 7	-	-	-	impossible since all 3 digits odd
3 5 8	-	-	-	combination impossible
3 5 9	-	-	-	impossible since all 3 digits odd
3 6 7	A053946	A053947	LINK 2 LINK 10	[17] 25226323103806424 2 = [33] 636367377337637773373677663667776 [19] 1932967502917049474 2 = [37] 3736363367333373666777367633763676676
3 6 8	A058465	A058466	LINK 2 LINK 10	[15] 183539278812156 2 = [29] 33686666866886336386333368336
3 6 9	A053948	A053949	LINK 2 LINK 10	[23] 18430047920827535573187 2 = [45] 339666666363999366939366969366969936633336969 [26] 31047286456844613647179386 2 = [51] 963933996333366963633963999333393696336393663336996
3 7 8	-	-	-	combination impossible
3 7 9	-	-	-	impossible since all 3 digits odd
3 8 9	A058467	A058468	LINK 2 LINK 10	[12] 199974958167 2 = [23] 39989983893893399999889
4 5 6	A030177	A030176	LINK 2 LINK 1 LINK 10	[17] 25425667278648884 2 = [33] 646464556564556546656456554445456 [21] 237605691124298293112 2 = [41] 56456464454655444665444666454556666644544 [23] 75269219840819770294592 2 = [46] 5665455455445656566644565645546455454464446464 [24] 675754811056988742949784 2 = [48] 456644564666666555445565455644644555565545646656 [26] 25781108305591628417975738 2 = [51] 664665545464645645665646644665564654546645556644644
4 5 7	A053950	A053951	LINK 2 LINK 10	[16] 8629863583949388 2 = [32] 74474545477575775744575745574544 [29] 27341447393189418631675507588 2 = [57] 747554745554544455555455747745774457774554454557445577744
4 5 8	A053952	A053953	LINK 2 LINK 10	[15] 767175898056538 2 = [30] 588558858558855585845444545444 [30] 293061185503724726684202222622 2 = [59] 85884858448848555885485448555548484845848584584884848554884
4 5 9	A053954	A053955	LINK 2 LINK 10	[24] 703957001491895099962643 2 = [48] 495555459949459999994555545994544549499995545449
4 6 7	A053956	A053957	LINK 2 LINK 10	[18] 253863101778786762 2 = [35] 64446474444746646444646744666444644 [24] 815277035409723028858892 2 = [48] 664676644466466777446466766667744446667647467664
4 6 8	A053958	A053959	LINK 2 LINK 10	[20] 26204321529981784378 2 = [39] 686666466846666884868486448488884846884
4 6 9	A053960	A053961	LINK 2 LINK 10	[21] 999997323321167445187 2 = [42] 999994646649499499946646996649944649464969 [23] 22236561446627600040614 2 = [45] 494464664969644944649644494946666694449496996 [25] 9718263579193026119075264 2 = [50] 94444646994669646646646969664664969996646496669696
4 7 8	A053962	A053963	LINK 2 LINK 10	[19] 9208064263934568938 2 = [38] 84788447488748874848488744487874447844 [20] 22019193553462506122 2 = [39] 484844884744844787448744774844887478884
4 7 9	A053964	A053965	LINK 2 LINK 10	[10] 8819171038 2 = [20] 77777777797497997444 [27] 865996535661545126193725357 2 = [54] 749949999777797799497449749797947997449449477944777449
4 8 9	A053966	A053967	LINK 2 LINK 10	One infinite pattern (6n)7 2 = (4n+1)(8n)9  [n⩾0] [10] 6670081667 2 = [20] 44489989444449498889 [26] 30732718558321504090886022 2 = [51] 944499989984998988844994898844848999494444994984484
5 6 7	A053968	A053969	LINK 2 LINK 10	[18] 815816631091556424 2 = [36] 665556775565576667756557666775667776 [21] 881854102200458483334 2 = [42] 777666657567776675657756675676567555755556 [22] 2562353735836753390526 2 = [43] 6565656667556566576676575665776756666556676
5 6 8			LINK 2 LINK 10	Only this one rather trivial solution is known [3] 816 2 = [6] 665856
5 6 9	A053970	A053971	LINK 2 LINK 10	[18] 834023722663550236 2 = [36] 695595569965566559565695699695655696
5 7 8	-	-	-	combination impossible
5 7 9	-	-	-	impossible since all 3 digits odd
5 8 9	A058469	A058470	LINK 2 LINK 10	[11] 92496431583 2 = [22] 8555589855588599885889
6 7 8			LINK 2 LINK 10	( No rootsolutions under 1025 info from Hisanori Mishima's website )
6 7 9	A053972	A053973	LINK 2 LINK 10	[16] 9831977256725526 2 = [32] 96667776776767999797799699976676
6 8 9	A053974	A053975	LINK 2 LINK 10	[22] 2588184048685235767383 2 = [43] 6698696669868698868988986669668968886668689
7 8 9	A058471	A058472	LINK 2 LINK 10	[16] 9949370777987917 2 = [32] 98989978877879888789778997998889

---

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 9/12/2008 20:37 ]

---

I found the following interesting infinite pattern for numbers m such that
the digits of m² are 0, 2 & 5 which isn't in the mentioned in the table

a(n) = 4.4.(9_2^1–1).4.(9_2^2–1).4.(9_2^3–1). ... .4.(9_2^n–1).5      [ n > 0 ]

b(n) = a(n)² = {20} . {20.5.(0_2^1–1)} . {20.5.(0_2^1–1).5.(0_2^2–1)} . ... . {20.5.(0_2^1–1).5.(0_2^2–1) . ... . 5.(0_2^n–1)} . {25}

Examples :

a(1) = 4.4.9.5 = 4495

b(1) = {20}.{2050}.{25} = 20205025

a(2) = 4.4.9.4.999.5 = 44949995

b(2) = {20}.{2050}.{20505000}.{25} =2020502050500025

a(3) = 4.4.9.4.999.4.9999999.5 = 4494999499999995

b(3) = {20}.{2050}.{20505000}.{2050500050000000}.{25} = 20205020505000205050005000000025

a(4) = 4.4.9.4.999.4.9999999.4.999999999999999.5 = 44949994999999949999999999999995

b(4) = {20}.{2050}.{20505000}.{2050500050000000}. {20505000500000005000000000000000}.{25}

        = 2020502050500020505000500000002050500050000000500000000000000025

I hope that the pattern and examples are clear.

Best wishes,
Farideh

---

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 16/12/2008 17:22 ]

---

I also found the following similar patterns a(n, k) for [ 1 < k < n+1 ].

 a(n, k) = 4.4.(9_2^1–1).4.(9_2^2–1).4.(9_2^3–1). ... .4.(9_2^n–1).4.(9_2^k–2).5      [1 < k < n+1]

The formula for b(n, k) = a(n, k)² is more intricate than of b(n).

I wrote the formula of b(n, 2) after the following examples :

a(2, 2) = 4.4.9.4.999.4.99.5 = 44949994995

b(2, 2) = 2020502050050525050025

a(3, 2) = 4.4.9.4.999.4.9999999.4.99.5 = 4494999499999994995

b(3, 2) = 20205020505000205005055005000025050025

a(3, 3) = 4.4.9.4.999.4.9999999.4.999999.5 = 44949994999999949999995

b(3, 3) = 2020502050500020505000050500052500000500000025

a(4, 2) = 4.4.9.4.999.4.9999999.4.9999999999999994.99.5

            = 44949994999999949999999999999994995

b(4, 2) = 2020502050500020505000500000002050050550050000500500000000000025050025

a(4, 3) = 4.4.9.4.999.4.9999999.4.9999999999999994.999999.5

            = 449499949999999499999999999999949999995

b(4, 3) = 202050205050002050500050000000205050000505000550000005000000002500000500000025

a(4, 4) = 4.4.9.4.999.4.9999999.4.9999999999999994.99999999999999.5

            = 44949994999999949999999999999994999999999999995

b(4, 4) = 2020502050500020505000500000002050500050000000050500050000000525000000000000050000000000000025

b(n, 2) for [ n > 2 ], includes 2n substrings and I separated them with "." and put them
between parentheses as follows.

 b(n, 2) =

{20205}.

            {(0_2^1–1).205}.

            {(0_2^1–1).5.(0_2^2–1).205}.

            {(0_2^1–1).5.(0_2^2–

1).5.(0_2^3–1).205}.

            ... .

            {(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1). ... .5.(0_{2^(n–1)–1}).205}.

{00505}.

            {5005.(0_2^3–4)}.

            {5005.(0_2^4–4)}.

            ...

            {5005.(0_2^n–4)}.

{25050025}

Examples :

b(3, 2) = a(3, 2)² =

{20205}.

            {(0_2^1–1).205}.

            {(0_2^1–1).5.(0_2^2–1).205}.

{00505}.

            {5005.(0_2^3–4)}.

{25050025}

=

20205.

            0205.

            0.5.000.205.

00505.

            5005.0000.

25050025.

= 20205020505000205005055005000025050025

b(4, 2) = a(4, 2)² =

{20205}.

            {(0_2^1–1).205}.

            {(0_2^1–1).5.(0_2^2–1).205}.

            {(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).205}.

{00505}.

            {5005.(0_2^3–4)}.

            {5005.(0_2^4–4)}.

{25050025}

=

20205.

            0205.

            0.5.000.205.

            0.5.000.5.0000000.205.

00505.

            5005.0000.

            5005.000000000000.

25050025

= 2020502050500020505000500000002050050550050000500500000000000025050025

Hope that the formula for b(n,2) and the examples are clear.

Best wishes,
Farideh

---

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ vr 19/12/2008 16:22 ]

---

I also found a very nice formula for b(n, k) :

b(n, k), [ 1 < k < n ] :

 b(n, k) =

{20}.

            {20.5.(0_2^1–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1).5.(0_2^5–1)}.

            ...

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1). ... .5.(0_{2^(n–1)–1})}.

{20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1). ... .5.(0_{2^(k–2)–1}).5.(0_2^(k–1)).

                        5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1). ... .5.(0_{2^(k–2)–1}).5.(0_{2^(k–1)–1}).5}.

            {5.(0_2^k–2).5.(0_{2^(k+1)–2^k})}.

            {5.(0_2^k–2).5.(0_{2^(k+2)–2^k})}.

            ...

            {5.(0_2^k–2).5.(0_2^n–2^k)}.

{25.(0_2^k–3).5.(0_2^k–2).25}

and

b(n, n), [ n > 2 ] :

 b(n, n) =

{20}.

            {20.5.(0_2^1–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1)}.

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1).5.(0_2^5–1)}.

            ...

            {20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1).5.(0_2^4–1). ... .5.(0_{2^(n–1)–1})}.

{20.5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1). ... .5.(0_{2^(n–2)–1}).5.(0_2^(n–1)).

                        5.(0_2^1–1).5.(0_2^2–1).5.(0_2^3–1). ... .5.(0_{2^(n–2)–1}).5.(0_{2^(n–1)–1}).5}.

{25.(0_2^n–3).5.(0_2^n–2).25}

---

Dear Farideh,

What intricate and beautiful patterns you discovered there! Thanks a lot !

---

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 24/02/2009 9:24 ]

---

Since the following six numbers are solutions of infinite patterns so they
must not be in the table of sporadic solutions.

1. 4495
2. 4949995
3. 4949994995
4. 4494999499499995
5. 4494999499999995
6. 4494999499999994995

The number 44949994999999949999995 isn't a solution of my infinite pattern.

Patrick replied :

But are you sure that the last number couldn't be made part of that
infinite pattern as well. It does look so similar to the others, perhaps
an adaptation 'somehow' of your pattern could include this one as well ?

---

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5) - New patterns
[ di 26/02/2009 4:29 ]

---

1.
For each natural number n, [ n > 1 ] there exist n numbers m of the following
form where m² has only three distinct digits 0, 2 & 5.

44.(9_2^1–1).4(9_2^2–1).4. ... (9_2^n–1).4.99.4.(9_k).5

Where k is in the set A(n) = {2^3–2, 2^4–2, ..., 2^n–2, 2^(n+1)–4, 2^(n+1)+2}.

Note that A(n) has n elements and A(2) has only the two last terms.

Since for [ n > 2 ] we have min(A(n)) = 6 and only for [ n = 2 ] min(A(n)) = 4,
I considered the two solutions related to A(2) = {4,10} namely 4494999499499995 &
4494999499499999999995 as two sporadic solutions.

But as you expected we can include these two solutions, specially the first one
which is correspondent to 4 ( the smallest term of A(2) ), in these infinite patterns.

Suppose that the i-th term (in increasing order) of A(n) = A(n,i) now we define :

c(n,i) = 44.(9_2^1–1).4(9_2^2–1).4. ... (9_2^n–1).4.99.4.(9_A(n,i)).5

and

cc(n,i) = c(n,i)².

Examples :

c(2,1) = 44.(9_2^1–1).4.(9_2^2–1).4.99.4.(9_A(2,1)).5

c(2,2) = 44.(9_2^1–1).4.(9_2^2–1).4.99.4.(9_A(2,2)).5

c(3,2) = 44.(9_2^1–1).4.(9_2^2–1).4.(9_2^3–1).4.99.4.(9_A(3,2)).5

A(2,1) = 2^(2+2)–4 = 4 ; A(2,2) = 2^(2+2)+2 = 10 ; A(3,2) = 2^(3+1)–4 = 12

Hence,

c(2,1) = 4494999499499995
c(2,2) = 4494999499499999999995
c(3,2) = 44949994999999949949999999999995

cc(2,1) = 20205020500505205550255005000025
cc(2,2) = 20205020500505250500205050005005000000000025
cc(3,2) = 2020502050500020500505500500002055502550000000500500000000000025

2.
For each natural number n, there exists a number d(n),

d(n) = 5.(0_{3*2^(n–1)–1}).5.(0_{3*2^(n–2)–1}). ... . 5.(0_3*2^0–1).49995505,

where dd(n) = d(n)² has only three distinct digits 0, 2 & 5.

Note that we have d(n+1) = 5.(0_3*2^n–1).d(n) and number of digits of d(n) equals to
8 + (3*2^0 + 3*2^1 + ... + 3*2^(n-1)) = 3*2^n + 5.

So for the sequence {d(n)} we obtain the following recursion relation.

d(1) = 50049995505, d(n+1) = d(n) + 5*10^(3*2^(n+1)+4).

Examples :

d(2) = 5.(0_3*2^1–1).5.(0_3*2^0–1).49995505 = 50000050049995505
dd(2) = 2500005005002055502050050520205025

d(3) = 5.(0_3*2^2–1).5.(0_3*2^1–1).5.(0_3*2^0–1).49995505
d(3) = 50000000000050000050049995505
dd(3) = 2500000000005000005005002050505005002055502050050520205025

d(4) = d(3) + 5*10^(3*2^4+4) = 50000000000050000050049995505 + 5*10^52
d(4) = 50000000000000000000000050000000000050000050049995505.

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Anne Zahn (email)
Reporting an error within a pattern of 0 1 8 .
[ di 12/6/2021 20:37 ]

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Hello,
I found a mistake in the table on this webpage. One of the infinite patterns is wrong.
For triple 0 1 8 there is the pattern 1(0_n)4(0_n)1 ² = 1(0_n)818(0_n)8(0_n)1 [n>=1] which is not right.
It should be 1(0_n)8(0_n-1)18(0_n)8(0_n)1 .

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Zhao Hui Du (email)
Update for squares containing at most three distinct digits
[ vr 1-2/3/2024 7:00 ]

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Hi Patrick,
  I have searched for the problem of squares containing at most three distinct digits
and found all results with length of base less than 27 digits.
  The results are available at a Chinese BBS (see LINK 10).

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I think we should thank the development of the IC industry for the achievement.
C++ code is used with 'openmp' to run in a machine with 40 cores to process one pattern
(square with given 3 digits) no more than 27 digits for its bases (so squares are no more than 54 digits)
The code first enumerates all integers so that the last 16 digits of its squares are in the pattern.
Next it enumerates squares whose first 16 digits are in the pattern and find square root of it.
Some digits of the two lists of integers should be overlapped for target integers so that we could
search in the first list for those integers with overlapped digits and verify the result.
The corresponding C++ code is in the BBS too.
https://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=19296&page=9#pid99352

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Zhao Hui Du (email)
Second Update
[ wo 6/3/2024 23:52 ]

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All results no more than 30 digits are available now such as
[30] 177324875114669443080086908188^2 = [59] 31444111334433114334141133143444444313434111431113141443344

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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