Sloane's On-Line Encyclopedia of Integer Sequences


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

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

012013014015016017018019024025
029034036039045046047048049056
059067069079089123124125126127
128129134136138145146147148149
156158167168169178189234235236
239245246247248249256257258259
267269279289345346347348349356
367368369389456457458459467468
469478479489567568569589678679
689789

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
TRIPLESROOTSQUARESOURCEPATTERNS & SPORADIC RECORDS (mostly L⩾17)
0 1 2A058411A058412 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 4A058413A058414 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 5A058415A058416 LINK 2

LINK 10
[17] 23452400954944999 2 =
[33] 550015110551505101015151115110001
0 1 6A058417A058418 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 7A058419A058420 LINK 2

LINK 10
[16] 8427200114569499 2 =
[32] 71017701771000177071770101111001
0 1 8A058421A058422 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 9A058473A058474 LINK 2

LINK 10
[20] 43694278824566964251 2 =
[40] 1909190001999001011109190090109911991001
[27] 100990098979999970099500001 2 =
[53] 10199000091990191001091091099001091999900190199000001
0 2 3---combination impossible
0 2 4A058423A058424 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 5A058425A058426 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 9A058427A058428 LINK 2


LINK 10
[17] 96385117673990673 2 =
[34] 9290090909029029202290909290992929
[18] 151400161921673747 2 =
[35] 22922009029909029220029029909020009
[21] 149067065510873088673 2 =
[41] 22220990020022929092929022220290920900929
0 3 4A058429A058430 LINK 2

LINK 10
[19] 5773251280200207952 2 =
[38] 33330430344333340030340440344044034304
[23] 20832739723817975138362 2 =
[45] 434003044400343443044430000434430333044043044
0 3 5---combination impossible
0 3 6A058431A058432 LINK 2

LINK 10
[17] 25107103902348156 2 =
[33] 630366666363306003336330636600336
0 3 7---combination impossible
0 3 8---combination impossible
0 3 9A058433A058434 LINK 2

LINK 10
[9] 969071253 2 =
[18] 939099093390990009
0 4 5A058435A058436 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 6A058437A058438 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 7A058439A058440 LINK 2

LINK 10
[16] 2010988315424552 2 =
[31] 4044074004774077447400004400704
0 4 8A058441A058442 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 9A058443A058444 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 6A058445A058446 LINK 2

LINK10
[16] 2236081408416666 2 =
[31] 5000060065066660656065066555556
0 5 7---combination impossible
0 5 8---combination impossible
0 5 9A058447A058448 LINK 2

LINK 10
[11] 70778174997 2 =
[22] 5009550055905955950009
[12] 771395165003 2 =
[24] 595050500590005595990009
0 6 7A058449A058450 LINK 2

LINK 10
[20] 26012881552428213576 2 =
[39] 676670006660660066767076066770670707776
0 6 8---combination impossible
0 6 9A058451A058452 LINK 2

LINK 10
[17] 30000101109940614 2 =
[33] 900006066606660060090966606696996
[29] 24691314243454114014126412353 2 =
[57] 609660999069000006699096996606066090996096009966990996609
0 7 8---combination impossible
0 7 9A058453A058454 LINK 2

LINK 10
[16] 8819172285373497 2 =
[32] 77777799799099990007000790009009
0 8 9A058455A058456 LINK 2

LINK 10
[15] 301345331969667 2 =
[29] 90809009099908808089808090889
[27] 299831600904572582192518303 2 =
[53] 89898988900998890088080098089989880890999988989999809
1 2 3A030175A030174 LINK 2

LINK 10
[20] 56843832676142723489 2 =
[40] 3231221313313311221231322223122312333121
[21] 557963558954625926861 2 =
[42] 311323333121312322332133323111223321313321
1 2 4A053880A053881 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 5A031153A031153 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 6A053882A053883 LINK 2

LINK 10
[20] 47130268582155593596 2 =
[40] 2221262216626122626662262611111116211216
[26] 51595698572871617009432954 2 =
[52] 2662116111222626216162621266666216222222216621166116
1 2 7A053884A053885 LINK 2

LINK 10
[18] 130834904430015239 2 =
[35] 17117772217211221211117217772227121
1 2 8A053886A053887 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 9A053888A053889 LINK 2

LINK 10
[17] 99961098929489127 2 =
[34] 9992221299191112291912929211222129
[18] 459556524411439511 2 =
[36] 211192199129121999192121999211919121
[27] 145303131149776986249167839 2 =
[53] 21112999921929291129929211912291229929191119991929921
1 3 4A053890A053891 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 6A053892A053893 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 8A053894A053895 LINK 2

LINK 10
[24] 286074095527510693610891 2 =
[47] 81838388131883313833383381811133318888113813881
1 3 9---impossible since all 3 digits odd
1 4 5A053896A053897 LINK 2

LINK 10
[14] 73560479506012 2 =
[28] 5411144145154411455544144144
1 4 6A027677A027676 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 7A053898A053899 LINK 2

LINK 10
[19] 1071033028175028538 2 =
[37] 1147111747441771474111741117114417444
[22] 2177492084289725902412 2 =
[43] 4741471777144414774147714111744447747417744
1 4 8A053900A053901 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 9A027675A006716 *
(* seq by
Neil Sloane)
LINK 2

LINK 9

LINK 4

LINK 10
[18] 648070211589107021 2 =
[36] 419994999149149944149149944191494441
1 5 6A053902A053903 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 8A053904A053905 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 7A053906A053907 LINK 2

LINK 10
[10] 1292931424 2 =
[19] 1671671667166667776
1 6 8A053908A053909 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 9A053910A053911 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 8A053912A053913 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 9A053914A053915 LINK 2

LINK 10
[16] 2969848344609859 2 =
[31] 8819999189981919818818919999881
2 3 4A053916A053917 LINK 2

LINK 10
[15] 205483392086668 2 =
[29] 42223424423443333243223342224
2 3 5A053918A053919 LINK 2

LINK 10
[18] 159789024443333515 2 =
[35] 25532532332552235533223325522255225
[27] 576387476638096486959455635 2 =
[54] 332222523225232223533222222253253255253352335533253225
2 3 6A058457A058458 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 9A053920A053921 LINK 2

LINK 10
[19] 1814641285211195673 2 =
[37] 3292922993992939999922923222293922929
[20] 14940646884386874573 2 =
[39] 223222929323939222223332232999233932329
2 4 5A031154A031152 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 6A053922A053923 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 7A058459A058460 LINK 2

LINK 10
[11] 88002411582 2 =
[22] 7744424444247727742724
[17] 47146022358675418 2 =
[34] 2222747424244722424422447477474724
2 4 8A027679A027678 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 9A053924A053925 LINK 2

LINK 10
[21] 222468490448488507807 2 =
[41] 49492229242429222429494944494949499949249
[21] 547259530974381470838 2 =
[42] 299492994242299992492444422992424244422244
2 5 6A030486A030484 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 7A030487A030485 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 8A053926A053927 LINK 2

LINK 10
[18] 159013392166264585 2 =
[35] 25285258888222255288288552225222225
2 5 9A053928A053929 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 7A058461A058462 LINK 2

LINK 10
[18] 150573163864701424 2 =
[35] 22672277676226226672276776667627776
2 6 8---combination impossible
2 6 9A053930A053931 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 9A053932A053933 LINK 2

LINK 10
[11] 14907304327 2 =
[21] 222227722297792922929
[27] 850281851974525726895170673 2 =
[54] 722979227797229279772792797979272222799797729799272929
2 8 9A053934A053935 LINK 2

LINK 10
[13] 5405829167667 2 =
[26] 29222988989999289998222889
[26] 17320185602062360469701767 2 =
[51] 299988829289888292222928892882898892999989922922289
3 4 5A053936A053937 LINK 2

LINK 10
[20] 21343231796858797962 2 =
[39] 455533543534444433554534333443535353444
3 4 6A053938A053939 LINK 2

LINK 10
[17] 66817996351009092 2 =
[34] 4464644636363464333646646666664464
[19] 6666680833328031344 2 =
[38] 44444633333463334436443663666646446336
3 4 7A058463A058464 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 8A053940A053941 LINK 2

LINK 10
[19] 6952948212333013522 2 =
[38] 48343488843384848488834343333834844484
[25] 5906300402396058566810062 2 =
[50] 34884384443343843348888488484833488344838384443844
3 4 9A053942A053943 LINK 2

LINK 10
[21] 185605616817891584607 2 =
[41] 34449444994349999433343349994949439344449
3 5 6A053944A053945 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 7A053946A053947 LINK 2

LINK 10
[17] 25226323103806424 2 =
[33] 636367377337637773373677663667776
[19] 1932967502917049474 2 =
[37] 3736363367333373666777367633763676676
3 6 8A058465A058466 LINK 2

LINK 10
[15] 183539278812156 2 =
[29] 33686666866886336386333368336
3 6 9A053948A053949 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 9A058467A058468 LINK 2

LINK 10
[12] 199974958167 2 =
[23] 39989983893893399999889
4 5 6A030177A030176 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 7A053950A053951 LINK 2

LINK 10
[16] 8629863583949388 2 =
[32] 74474545477575775744575745574544
[29] 27341447393189418631675507588 2 =
[57] 747554745554544455555455747745774457774554454557445577744
4 5 8A053952A053953 LINK 2

LINK 10
[15] 767175898056538 2 =
[30] 588558858558855585845444545444
[30] 293061185503724726684202222622 2 =
[59] 85884858448848555885485448555548484845848584584884848554884
4 5 9A053954A053955 LINK 2

LINK 10
[24] 703957001491895099962643 2 =
[48] 495555459949459999994555545994544549499995545449
4 6 7A053956A053957 LINK 2

LINK 10
[18] 253863101778786762 2 =
[35] 64446474444746646444646744666444644
[24] 815277035409723028858892 2 =
[48] 664676644466466777446466766667744446667647467664
4 6 8A053958A053959 LINK 2

LINK 10
[20] 26204321529981784378 2 =
[39] 686666466846666884868486448488884846884
4 6 9A053960A053961 LINK 2


LINK 10
[21] 999997323321167445187 2 =
[42] 999994646649499499946646996649944649464969
[23] 22236561446627600040614 2 =
[45] 494464664969644944649644494946666694449496996
[25] 9718263579193026119075264 2 =
[50] 94444646994669646646646969664664969996646496669696
4 7 8A053962A053963 LINK 2

LINK 10
[19] 9208064263934568938 2 =
[38] 84788447488748874848488744487874447844
[20] 22019193553462506122 2 =
[39] 484844884744844787448744774844887478884
4 7 9A053964A053965 LINK 2

LINK 10
[10] 8819171038 2 =
[20] 77777777797497997444
[27] 865996535661545126193725357 2 =
[54] 749949999777797799497449749797947997449449477944777449
4 8 9A053966A053967 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 7A053968A053969 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 9A053970A053971 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 9A058469A058470 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 9A053972A053973 LINK 2

LINK 10
[16] 9831977256725526 2 =
[32] 96667776776767999797799699976676
6 8 9A053974A053975 LINK 2

LINK 10
[22] 2588184048685235767383 2 =
[43] 6698696669868698868988986669668968886668689
7 8 9A058471A058472 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.(92^1–1).4.(92^2–1).4.(92^3–1). ... .4.(92^n–1).5      [ n > 0 ]

b(n) = a(n)2 = {20} . {20.5.(02^1–1)} . {20.5.(02^1–1).5.(02^2–1)} . ... . {20.5.(02^1–1).5.(02^2–1) . ... . 5.(02^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.(92^1–1).4.(92^2–1).4.(92^3–1). ... .4.(92^n–1).4.(92^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}.
            {(02^1–1).205}.
            {(02^1–1).5.(02^2–1).205}.
            {(02^1–1).5.(02^2
1).5.(02^3–1).205}.
            ... .
            {(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–1)–1).205}.
{00505}.
            {5005.(02^3–4)}.
            {5005.(02^4–4)}.
            ...
            {5005.(02^n–4)}.
{25050025}

Examples :

b(3, 2) = a(3, 2)2 =

{20205}.
            {(02^1–1).205}.
            {(02^1–1).5.(02^2–1).205}.
{00505}.
            {5005.(02^3–4)}.
{25050025}

=
20205.
            0205.
            0.5.000.205.
00505.
            5005.0000.
25050025.

= 20205020505000205005055005000025050025

b(4, 2) = a(4, 2)2 =

{20205}.
            {(02^1–1).205}.
            {(02^1–1).5.(02^2–1).205}.
            {(02^1–1).5.(02^2–1).5.(02^3–1).205}.
{00505}.
            {5005.(02^3–4)}.
            {5005.(02^4–4)}.
{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.(02^1–1)}.
            {20.5.(02^1–1).5.(02^2–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1).5.(02^5–1)}.
            ...
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1). ... .5.(02^(n–1)–1)}.

{20.5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(k–2)–1).5.(02^(k–1)).
                        5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(k–2)–1).5.(02^(k–1)–1).5}.

            {5.(02^k–2).5.(02^(k+1)–2^k)}.
            {5.(02^k–2).5.(02^(k+2)–2^k)}.
            ...
            {5.(02^k–2).5.(02^n–2^k)}.

{25.(02^k–3).5.(02^k–2).25}

and

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

 b(n, n) =

{20}.

            {20.5.(02^1–1)}.
            {20.5.(02^1–1).5.(02^2–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1)}.
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1).5.(02^5–1)}.
            ...
            {20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1). ... .5.(02^(n–1)–1)}.

{20.5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–2)–1).5.(02^(n–1)).
                        5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–2)–1).5.(02^(n–1)–1).5}.

{25.(02^n–3).5.(02^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.(92^1–1).4(92^2–1).4. ... (92^n–1).4.99.4.(9k).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.(92^1–1).4(92^2–1).4. ... (92^n–1).4.99.4.(9A(n,i)).5
and
cc(n,i) = c(n,i)2.

Examples :

c(2,1) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,1)).5

c(2,2) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,2)).5

c(3,2) = 44.(92^1–1).4.(92^2–1).4.(92^3–1).4.99.4.(9A(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.(03*2^(n–1)–1).5.(03*2^(n–2)–1). ... . 5.(03*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.(03*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.(03*2^1–1).5.(03*2^0–1).49995505 = 50000050049995505
dd(2) = 2500005005002055502050050520205025

d(3) = 5.(03*2^2–1).5.(03*2^1–1).5.(03*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.


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

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(0n)4(0n)1 2 = 1(0n)818(0n)8(0n)1 [n>=1] which is not right.
It should be 1(0n)8(0n-1)18(0n)8(0n)1 .


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

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).

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

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

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








 

[up TOP OF PAGE]


[ ( 1996 - 2024 ) ] ( © All rights reserved ) - Last modified : March 8, 2024.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com