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[ July 15, 2015 ]
Finding a pandigital as a substring in the decimal expansion
of that same pandigital raised to a power
Here I am looking for pandigitals raised to a power p so that the same
pandigital pops up as a substring in the decimal expansion of that number.
For that purpose I use UBASIC. One limitation here is that for the largest
pandigital an overflow occurs when p is greater than 107. Someone who
is equipped with better tools can raise that exponent to higher values and,
no doubt, will certainly find much more solutions. Good hunting! P@rick.
The zeroless pandigital version of this topic can be found
in my dedicated webpage on ninedigits
1804562739 92 =
38567606378153386362543294503351230718719686692921599700831300135786681309364
97256938032021568494256035243834608271977013121190984484541252395770975909951
34201360686826435136805102500052406807578382695453478267654927523758050000659
94323655829797003213842672722184235962032267730819453834571571062235329216095
97523784323618216170730514884517573667894774860430797939859821129039236430714
45816173536000968131315066441910994981522537859869257076784748797864603510403
62745564930002536001133095630951180456273965881889903374377040635650755805432
54893666723140904681099374249325973140310113614617111790959898098573828942281
69083487855584795335447997247875138356437525026482022256797814050887566658518
12707723719852966043494169765536560105604856707567859283208510952356345479069
97546068483988528230748173807857541360826428717200205481952796044514600437475
21521
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3416295807 41 =
75123218303037031434183531026708521916479271517384564516334017358436528904433
68372173821385119789342443919067383288849733441200193903775382036235147125218
95079843624682239238614037530392004359811356469822606518562344516012802556858
76958101412227791231919020473597496955538485781645038952336857129543445332630
91762441257215102641300560421775608182100160409175883159019810334869856873416
295807
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3416295807 81 =
16519347992765188675704997366174250964546241929455734027491010120826507343526
69132902186817378922555149361362451444441161177387947296006920063880631073287
43512401047146839914984743208962682354379674021323558975721277849138439129727
20283836828766224186654867794738477356665994995733899389835540478986841515901
80604436837708531182503034745314724853721892194304194745099974084503392403712
66807373238258882107763709931959929556651364515810680542831698334585084576223
49401177919962118819056368288961549884331571973695819358487954842111024619387
00547266753102476900366892634778521649458331507575034685814557308209879661695
02022426667464740279180842586777655670877643572678454009180282502705933552786
95556793580846172363666425481438316229672318157228136765097369739713743416295
807
In both cases the expansion ends with our pandigital !!
In fact I detected a pattern here as for all exponents +40 (starting with 1) it occurs.
1, 41, 81, 121, 161, ...
Can this phenomenon be explained mathematically ?
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3657812490 51 =
53002135104434083974295107729733667717040654846476097425485261220328866855852
71504803441128772764357739017148646788411029412276275042922694892434396254143
48793515708522071862882724102364998434235736846330684845034104402120935766976
01022971282572645737276412472217099390944995575161648761003059841167281107930
16137006513587023917737671292908464672118160331833046277362244795625534877968
63689076758062177287118288369443992840081973657812490000000000000000000000000
00000000000000000000000000
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3657812490 101 =
76800719919590078432925071974148597876877854062105916684763833185937781705753
83281476183683178644147316682717375665654255225060926550994397177085801874805
26553609913039442760221214744703267598894728210679926761402337567959581385266
21082622518065798159170419928139324519300943918848638348519415234032122155244
05988689600296989099668088818911186434860869532144452751639276824763624763754
04920817120720345528595874747470398626320900446802670284758650353286638757355
75582076312539413336473920504799681382640754693048217806233753987807634460979
23191578353228051041430597711168265165107287178147986560086750154917731725313
59086749279932313619464359933606406209548895901679912639907567337083903120432
99869965011658567583696528762695114783608021250377198923581674468638379220281
62192471621988626202321698325792844640652321364291819301555023466728895229239
56801639436578124900000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000
In both cases the expansion ends with our pandigital before the string of zeros !!
Another pattern surfaces as for all exponents +50 (starting with 1) it occurs.
1, 51, 101, 151, 201, ...
Can this phenomenon be explained mathematically ?
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3690812754 63 =
53541039610060376851011097630412065427904312359133126224509693438202295195314
46738815614019269902500266665191040908622671635043504944615701014394575688128
02379615286110348521914158153221541338155893204549200760125297159007406628165
63443292052751824472980063876586730946558509565184488024436834957582589645299
90532336908127544238807293002194754722385663262657761420123313442631219226967
54430185156642296899923332512422248773219331990496105403644866003832714644779
51523243348029085827931098616895782210529715262064942719981964637345205098143
8072485834382218261679268315932528447702856549150024947271204864
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4290185763 53 =
33207268890604080135287733076741093295523147901486740288913579418944681293352
51453488700040417609336160572419844214890993170836287298407254520417782825588
08685117150657321020388179127570169156593970366428459852395997205112185811281
08862328691308906810773908789440426301244381944220589259935826842901857635006
61221446895031486106603039713903349204569566862680593962765512891792960999040
52631974261502273229558132850378043137599648385001750862251131114929431465668
9466806082611936238585356680242037983206275694803
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5079186432 101 =
19284551719834380009459480372566211600627155742718929307674519952831402715430
43291537079427539164462823353105396873314162134000018002062442554515075758325
75044586228411476348215368706342908923003157882727402172459095313762612435420
32401420786422366888757376309871286671145566500609086164515495935427066833943
06488112753109364419219402904154418147214734152233936208881948974477548388640
95240911514975181347824610931393216556442820625395816515528706283153767687519
36464231059400755933542734248070153335291459282376331667172615278369991678999
29759859222955488781241331934265928935712926702104350076502480384065157905616
89629854812536324602258976312592774297642142473599223146954256250775898126029
28023558821852105897452394311872766247187205867492908114301784932357632265923
59270518863535802672594460550325375043929759509006841383470763775728701437998
38319448901571334160494981539263260402737906322598121394957839844220479796502
665932686781968112247764118837478535516786931775079186432
Here again the expansion ends with our pandigital !!
The pattern goes up for all exponents in steps of +100 (starting with 1) like
1, 101, 201, 301, 401, ...
Can this phenomenon be explained mathematically ?
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5340972861 66 =
10542521594777346143525866181010137876125249045799489387195032222610670437111
97989550047945993035947949172672351098651277122325946281719864886511503453556
12684037973114404361685543130706932526953905353730688655884597879890927351845
37747855157780170103004511779335138725287851869710006474680739400622876318630
14790595890002792952085790174938316794789654430404948289854213958682288290846
77005265432805331000574547585228672227991749428376748048143288384960755314554
49966280811687125340972861766291270941538637597588647910545114698925549530631
53259672080535686788269298917315758377331020157476940940105993821503953646958
545369164081039531342010761
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5734081269 93 =
34445895734081269299282235616926677882491979077695082589502616075210949130441
87695630219298364899979722619709531342573366406649920712306632165238073062266
82665647213451277075978052235753201794390508517266508632803572925734499868429
23476958006968704413733042369962168808616971634948749920909293774013449660178
52882407117769387144830205429270933208388521671499095447964489598703288896983
53803768437928632790161499661758491445342111359757762320454233419923797849515
63485028050598284082639845418159548976050565170851616897121266947693548642778
48160201022722566008471816830907291215931856275828297140621174892502419416340
30661072304955783678088274669580661303958882315925232161644006218794638101402
78766936848584314557889760304254776576301624701850533850707193914164059885404
26580825487701739170832341736182495037240794067901608498624210098683700429956
1512766523266031961734761839339268231668430660253331565099909
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5801934267 88 =
15646123925025288776728085019967857289037726797245099104166226030835688818953
39799153419312974195827316152025925730022578935049512759506126605057777772117
23599049499688125681486934142071360742504978424755879385583743311813837994798
05250961582841157498931253714021445263919102846642141213520892826121935641518
93487102031309653706475143495620622820357958019342672813117010675366006791403
93777695563828515139041409663374180805737357921221388056192746831223145943107
95689347894975104080064369151827172676090712813403976272817755267210965154543
00925395139278891553550387072191452393597585949608202380721825650301810959254
11849282176142528581156638461811081259450353695340560536636681964793294881812
09280398328024864490316710051224588423236249386486401649234432243681745180590
34861488382059453817286927802661392161411709372119378547379695737603242461438
6661458311841
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6357812490 101 =
13616930906420442043259405929999866138890181953431470395921699407085117760783
79536986744241462996090516980892457600810938519412701377694639547895160589971
79940437644604969159226080307904671668397831678693666758678744510837077805030
78206504513344786576043188805369174334775868464100758383598453386529564392290
31367832734849277419826252716936311745138781812648300098007290980852223803079
72137771591659178231093099603625082034017722405516753317039980819246827005117
36976451379012331388480253663967821868072501222801850425960780267385323754435
07539416471208237069631672456024450838621340577834346701338003303401309670932
01973176630643853667096489340849126751031165929265273487122455492177758908045
89274046128442756534554210620839547826578936054888844310552044248918565678393
81374739820431481832250271626095365734918171778895773396091917556338935202137
30494104101072263086569632930164216357812490000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000
|
In both cases the expansion ends with our pandigital before the string of zeros !!
Another pattern surfaces as for all exponents +100 (starting with 1) it occurs.
1, 101, 201, 301, ...
Can this phenomenon be explained mathematically ?
|
6359807241 47 =
57791741756534722727004392906529399079359807065576784033354898108479782887544
57980708766387168903254535742667107578663107040344190299065688019163630730729
65232756049512984865466510764939095635980724123613472190223429748689384471035
78815009439339180597621700663675268912571993456988884184091807134053221584001
29130573057784113904868246898385626782595973073916100755255450086114546267674
4538580359865793664294920600546382134145582561814542777036868228938362525881
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6384915702 88 =
71392857538052784902588115256925753557462423869196533512385589018764449416434
58924232319803741730633296088951077698317706571647389961240298916689359467436
56928121363990560013385531193127471649800268250313153559584901454387847517200
55293450661162773761440729301462554137051198234280824378472406356927494065217
78674070557400561950542351342494027665680749036921803550590345644754461059129
75006444720169391242713028750326432144829589140792928559036363144778900445920
86879348464093244577434613065926736047883779119721895725690744772607374971271
38906237693809776421163849157022551874412080709528675849109607617062448929078
76916445541588877299158159911956826315196248505439876523595046618624353488960
40267516165704849266200086623375209339391175450352760694915384184921992923975
42796322156802930596586623337290823061815608759526202674156436754375500538346
7199187549945856
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6957804213 74 =
22026377982394663972791178157367267188570188592261092581724744371815679604290
24738458945176376016136568069806790265044130458043828607415409299848147414533
54261031561350398095773580622231617754098541569347894435645046841327740422597
69770308413573036963744821913774642313848681273913276402285769968243061753707
41019945733551609332425932598399668699410556487633141170620867368503439480600
03154546460575831669070562965982715490780884942606227344836289858122942245246
96738360357264935312759983507977336741439695780421392739668293251325184323235
44096357102827459077235951568701851364751059690031039767400543329754267083975
35448081240567865309439439380774841598262758405526841689244519227737539192497
822969459506313843878205819463251289
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7405963182 90 =
18295293461335003487302810163041707869243658852161225533260633726799214855312
82662669355708128979465821899045105325664671924308236374059631829793893540429
14190359433808513331266358095609658889615767635847337446240659448909081637351
79035518152881734717021224290359666041493059231523087308727794184691188342133
67591167670351518564462513261348289382256744541925881579269501379381063522603
72003071465240902199959658934781035750641977367465935469749680424194290375235
06556758463446511610758017224523661426564615700793850991450060196119689952659
86571574790163157023188329197343323560533817371376390305601508128778819251761
99327396199169340921883927678604976790851613003932107001237835834006225291424
91774670361653113960164457432123217187970224085110883805761063972015651824082
39498465804841876093744099106495901190980556064351438422982640603552678033307
805911046836729149103013437684034907930624
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7502318469 71 =
13768296131429638491482483524897969805942306129484700510731148351303406544944
76294147301822455093388382344645767783831924410861515587596339065706636452913
16190604433953794802857865810632329994595173648759732928021724213088781857647
64180657514817621659610168726061907648035430121574150621534746072907211096304
85752728649860122745261144939463578527502318469831321901762307895804139600438
54938706938735061462188856253220664587065284277863259020255704310298839203175
45920608012533136732256782606397192947700164460045332505635184921397632148565
89841711178991369202569847859483641937056795523709619805567316561335467147142
64079131834842929598971675368998351005368040042353513359223647480225851092181
816989869
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7591864320 101 =
82260666872235555762250266625924353034740692041723072613225221908129972517410
79368040497793129185854340245930758942254133210108110971390842331027330555511
45540952916667198724287767754930598868574123864483173835155518388996830584546
13989804825083382539061412326421979239699525638449980690283152776976732040942
57670172876623812592321940729721672348124710665628420292270901806076777682020
97119707042032611205871824644392576564139193660812280032933061445605398390813
62055942901999401281725178269911388104228461256806247365431808150705536156893
18074291525522027694579819766996084171946251019573016891903101807872335414382
00201049471880295553020076943765175630219726442008847791105863542328160017318
83759251038778125207733504866110163801274171321479184174034631872273411246535
87680690571780670261116973775768781659632439322592913922927917692091904847243
44882494258519860070773612938144395368838759186432000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000
Here again the expansion ends with our pandigital before the string of zeros !!
The pattern goes up for all exponents in steps of +100 (starting with 1) like
1, 101, 201, 301, 401, ...
Can this phenomenon be explained mathematically ?
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7895612034 86 =
14968518741179234011119986977949117805974069214610754868018801800123476873765
37433027348589192071488850455714796357930121564435287576428136313650052867092
50321128959039963553997604035666993036955503107886815036971794758000252769450
89486797493254402637985675095284032754666094443700510351138468985801579256443
17652213338602359225117085190153735440999552372778294554786463969301291962997
35692130725826514960004855244978582789561203402375859920595500346845985281354
01699412523527467764299724097440301009496594067518042071462345106714853150356
20206070573444006117378111766207707101284765997771775423253468208619593357114
19815661838730126516954525809141537879070800930883021708980083836629157793907
37467688706278952011599279222160734410180347801619280744385994883833039651839
90034119572584840375086588829124120745807692767044880516667458010427654158847
83616
|
|
8504327169 92 =
33641571651290793860212391520008751940373515004796582698108323779816144435344
35910117988652146708363862598199573377563687102135595903551797194065272032020
81680878225253079722572167499574458377842289337040083969781457264006902467646
58296504457071007884970987303670910290946621457105878307149850432716981518170
93877027678486059032127432171890168864276396600164942107521557398655737319672
07683163286117672260865851328643427788830652601928902650088756773837770144496
03575247653746968948652719486078586105029078248712032729379488795262050831555
83615417762197170030015828211197971814538133288478758629296612572928918737044
96891797590925550405282514880333144047845999903611323091364185119881396498104
03524140542464350650269646769581916871610324166250869143140163359281995288713
88828013955192494451674875804568712857690992495722124273966164688611865332948
4957179589962763639380318376638926387233905113092133341271851765761
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8509342671 40 =
15697862526350727125934548496501997695628059847756287996202935065643014039084
30847541103760993858708589371810611306156300220319635303287838265753140201465
15870914690119122389819545902969541487271685655793973736526019623683175816207
52515885093426719914044690242024476604364544314646150474851508588871241106519
22588815609453939522083531453764426812992718022636258471065902838985825003234
2013320188801
|
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With 8509342671 I found the largest pandigital with p ⩽ 107 !
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A000195 