World!Of Numbers		HOME plate WON |
The Nine Digits Page 5 with some Ten Digits (pandigital) exceptions
Page 1 Page 2 Page 3 Page 4 Page 6 Page 7

When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

[ July 15, 2015 ]
Finding a ninedigital as a substring in the decimal expansion
of that same ninedigital raised to a power p

See WONplate 195 for the pandigital version of this topic.

125387649 116 = 24977680256511635554308807465784640450873233766747803697457826665337615090115 65231746558064246784915939331799792083405477295137019822599431230243339336256 71036099290638462808759659688825404030171009447306472501571623792897838640534 77085356208869947832016110312264802555478542369431139240340476254337276270686 34662823179078224977906450230607287868664539203911690841768842604378676955543 90100851394887980264901371704943840980429647269197342633254961700394120603361 43842807514599611962386883891826228850335619243196918128762543471635650559682 38318742855037099538294619343456338486825737070338628267534489179698733254887 09140884645181171985004192856213046111681831479406534681094572418253243315232 83754693737121802777642827488524555131566356534189708692077962292682715962963 10951543077827723454799874681321232284394839374645645842281885435452335843550 95833227039437289409212070147169984083138099798709210400021021253876491425914 0643857408857601

217653498 73 = 45382931250436248017939755904905375074156752993340811447973427529953789857607 32955682451554486827163229743678033027505512286752585680945126547840935996116 54815664694471438604641495185180470978832928274085612674105757197078023302686 63726802021863993886105320204264424109074548867821601860831321946603899915989 29493904262834904701039108240350414664604821797810564424853529625576570072775 66618409001769861258075239997032601466426217653498522505043608739599041141286 06969595596576995845246008728621299765577774490078132467180959116981633437817 8405015126965623686132913457323839428590795742775207720401436156100608

365127984 56 = 31399785292188574093400788496274173401686979859312394666521792490946925319997 76524136398355426736199589711873651279841242087305377690869763740464625716529 59057594291916954556200100043602722007077744099905403348921821982324181941209 69999260251709925043458412798095491979509046250663030898453529337668960610349 03081409266580407331283046683054359019742088912241401726530525075638795375443 40762300436359126476728589070428637730840695632269631597955668180598026800883 697316427481481216

365781249 51 = 53002135104434083974295107729733667717040654846476097425485261220328866855852 71504803441128772764357739017148646788411029412276275042922694892434396254143 48793515708522071862882724102364998434235736846330684845034104402120935766976 01022971282572645737276412472217099390944995575161648761003059841167281107930 16137006513587023917737671292908464672118160331833046277362244795625534877968 6368907675806217728711828836944399284008197365781249 365781249 101 = 76800719919590078432925071974148597876877854062105916684763833185937781705753 83281476183683178644147316682717375665654255225060926550994397177085801874805 26553609913039442760221214744703267598894728210679926761402337567959581385266 21082622518065798159170419928139324519300943918848638348519415234032122155244 05988689600296989099668088818911186434860869532144452751639276824763624763754 04920817120720345528595874747470398626320900446802670284758650353286638757355 75582076312539413336473920504799681382640754693048217806233753987807634460979 23191578353228051041430597711168265165107287178147986560086750154917731725313 59086749279932313619464359933606406209548895901679912639907567337083903120432 99869965011658567583696528762695114783608021250377198923581674468638379220281 62192471621988626202321698325792844640652321364291819301555023466728895229239 568016394365781249 In both cases the expansion ends with our ninedigital !! In fact I detected a pattern here as for all exponents +50 (starting with 1) it occurs. 1, 51, 101, 151, 201, ... Can this phenomenon be explained mathematically ? Yes, as Alexandru Petrescu does with an elegant proof hereunder.

Proof By Alexandru Petrescu (PhD in applied mathematics) [ April 17, 2022 ] Let a = 365781249. Condition is: 109 | (a51–a) or 109 | a(a50–1). But gcd(a,109) = 1 so 109 | (a50–1). Factorization: a50–1 = (a–1)(a+1)(a4–a3+a2–a+1)(a4+a3+a2+a+1)(a20–a15+a10–a5+1)(a20+a15+a10+a5+1) a–1 = 365781248 = 28 x 7 x 17 x 12007 (1) a+1 = 365781250 = 2 x 57 x 2341 (2) For any number b having the units digit equal to 9 we have: b2k = 1 (mod 10) and b2k+1 = 9 (mod 10) So: a4–a3+a2–a+1 = 1–9+1–9+1 = 5 (mod 10) (3) a20–a15+a10–a5+1 = 1–9+1–9+1 = 5 (mod 10) (4) From (1)-(4) we have 29 x 59 = 109 | (a50–1). Generally 109 | (a50p+1–a) because a50p–1 = (a50)p–1 = (a50–1)(....) The vertical bar "|" stands for 'divides' Pari/gp has a simple command for producing the 6 terms of the factorization of our polynomial (14:00) gp > factor(a^50-1) %1 = [ a - 1 1] [ a + 1 1] [ a^4 - a^3 + a^2 - a + 1 1] [ a^4 + a^3 + a^2 + a + 1 1] [a^20 - a^15 + a^10 - a^5 + 1 1] [a^20 + a^15 + a^10 + a^5 + 1 1]

495126738 61 = 23861578656021269346703174520372315791816251470087567211478523202968124667924 93848688084609784146764549823116272246105113757348275839028643583491902670379 64611114317468350096247548658050991212656177878337691054141810387339185188226 37443389139903195571962109654889436933754829870083357415260689569881177088929 20725614584762139979825699821114369929192812850477252290195382123150020328432 81324949512673878966294790960293349542846852039693433215569689109114314341111 102344236031307150080447721778461888506037971062682424038575401074688

623479581 83 = 93378199229292611484783069958832018204031360211962347958195478201099393097916 34096128381183555954877721494796498660605839598106441880144452469796271440667 05689066988711351184847509304365224018029344080389906241440975034338043383492 65523244487441047103333380133096323111353945707309764846648779004687899784490 95209825853284627548178988792267472945800848267750379848985027317431251568478 10303914183404736553013121265261006876701288947711222743493437503622840039296 31760526249913511031756985513898026705212045956997380233251890795219842954963 98174803095091720475573906927745701371461428793178627967034408092869119201862 69304525412368590073394987109053612598099858622032202094910455231063705993302 5724027684578757254054582767435966341

634812579 98 = 45631141858726733466372929460343476150274704363477974505922266227178826889112 03296142625749780118820787364378776141951859373704464157741137151720607037161 59124655023048774995732874441504376314578129603660792575268298258639078264194 85376075066280259241460659377367254814707313855803853251377152515963933782042 82013024297285193725035357126051237614145162756780783309796660052474365080592 57486561925965907731081653418482644904546883539412866892308710063071186726094 04162947833101908631744436257625261404831947239275944162745006380618099565297 96459806577080966775040859019302889640794901252275276440231781404017352377903 60488874615440260964529302179978922299373037300601770829712001588815559136063 48125792841936687381899103964185439346538524710921313754184454000657029185913 07178775895169595052782549109915160489328969529916734441976537436220053655780 8083316053484361

635781249 101 = 13616930906420442043259405929999866138890181953431470395921699407085117760783 79536986744241462996090516980892457600810938519412701377694639547895160589971 79940437644604969159226080307904671668397831678693666758678744510837077805030 78206504513344786576043188805369174334775868464100758383598453386529564392290 31367832734849277419826252716936311745138781812648300098007290980852223803079 72137771591659178231093099603625082034017722405516753317039980819246827005117 36976451379012331388480253663967821868072501222801850425960780267385323754435 07539416471208237069631672456024450838621340577834346701338003303401309670932 01973176630643853667096489340849126751031165929265273487122455492177758908045 89274046128442756534554210620839547826578936054888844310552044248918565678393 81374739820431481832250271626095365734918171778895773396091917556338935202137 3049410410107226308656963293016421635781249 635781249 201 = 29164245973260567384593998725899801391036529801483811790123121455125852321698 22356483494243511951735290740862875830839305968363755966766223109083382183664 88576657374238845416864831751683511906055638926371154956341508787131097544659 79522044789543649215998567411972121286520461446088073616511904873309159121071 74684934306957468357280843825646472100322416427703819568376723780664487397781 36609640440854544270101742492038697981640355680661735059638023384998850172032 66356797551332806280817340906746100461811475325319411146411680855221735010791 53992862953768970337009000514903631071241106546078919582643227455640762805923 03709660963193290082477816666688811748112154031780276816328007053689328670077 19486369454747397333719686514715621111571267203548578415858045081152038488373 19738994264925412594037600610232404954957253834070105970798906085984928801129 90069112802435045442019014381173623658922975667291863467667559295986468544645 95871328306360393761967216238027050306942300675363534043656652175578719662302 89488669296643593144384841208946918403617436312679072638091933929765488568810 57706701231797058492686450589593235151332428321380402982579558838646689565348 91937561213424352699023041796996956645347622484730341432324511585887735175214 61796191282018130190838622236792597334721732103343855693075534342114447377455 01930307417746262714112216038771957185176456937958730457766096901663183790850 21632420423456465784563445754553686156000076963413532065706676351187746174688 24823883482637094613008485441319259453071734490504331011445956738207887289935 52707227273816058424570137298617616776868455983820558158072745229332917127873 20544768174850588910654282156308815210119182479471883285722869803551761642790 5893653672809004042634257177233841040848855367748426195935586032842635781249 In both cases the expansion ends with our ninedigital !! Here again I detected a pattern as for all exponents +100 (starting with 1) it occurs. 1, 101, 201, 301, ... Can this phenomenon be explained mathematically ? Yes, as Alexandru Petrescu does with an elegant proof hereunder.

Proof By Alexandru Petrescu (PhD in applied mathematics) [ April 18, 2022 ] Let a = 635781249. Condition is: 109 | (a101–a) or 109 | a(a100–1). But gcd(a,109) = 1 so 109 | (a100–1). Factorization: a100–1 = (a–1)(a+1)(a2+1)(a4–a3+a2–a+1)(a4+a3+a2+a+1)(a8–a6+a4–a2+1)(a20–a15+a10–a5+1)(a20+a15+a10+a5+1)(a40–a30+a20–a10+1) a–1 = 635781248 = 27 x 4967041 (1) a+1 = 635781250 = 2 x 57 x 13 x 313 (2) For any number b having the units digit equal to 9 we have: b2k = 1 (mod 10) and b2k+1 = 9 (mod 10) So: a4–a3+a2–a+1 = 1–9+1–9+1 = 5 (mod 10) (3) a20–a15+a10–a5+1 = 1–9+1–9+1 = 5 (mod 10) (4) 2 | (a2+1) (5) From (1)-(5) we have 29 x 59 = 109 | (a100–1). Generally 109 | (a100p+1–a) because a100p–1 = (a100)p–1 = (a100–1)(....) The vertical bar "|" stands for 'divides' Pari/gp has a simple command for producing the 9 terms of the factorization of our polynomial (14:00) gp > factor(a^100-1) %1 = [ a - 1 1] [ a + 1 1] [ a^2 + 1 1] [ a^4 - a^3 + a^2 - a + 1 1] [ a^4 + a^3 + a^2 + a + 1 1] [ a^8 - a^6 + a^4 - a^2 + 1 1] [ a^20 - a^15 + a^10 - a^5 + 1 1] [ a^20 + a^15 + a^10 + a^5 + 1 1] [a^40 - a^30 + a^20 - a^10 + 1 1]

759186432 101 = 82260666872235555762250266625924353034740692041723072613225221908129972517410 79368040497793129185854340245930758942254133210108110971390842331027330555511 45540952916667198724287767754930598868574123864483173835155518388996830584546 13989804825083382539061412326421979239699525638449980690283152776976732040942 57670172876623812592321940729721672348124710665628420292270901806076777682020 97119707042032611205871824644392576564139193660812280032933061445605398390813 62055942901999401281725178269911388104228461256806247365431808150705536156893 18074291525522027694579819766996084171946251019573016891903101807872335414382 00201049471880295553020076943765175630219726442008847791105863542328160017318 83759251038778125207733504866110163801274171321479184174034631872273411246535 87680690571780670261116973775768781659632439322592913922927917692091904847243 44882494258519860070773612938144395368838759186432 759186432 201 = 89132484842206877224072333243882506481115047313501101593088591184654074856787 63195684432935384031288734506632618449021116516970989054557921092003073207419 78479610779365972342221373501421522475350316784210568722785016071585147887492 02734539633763612017476752173919263236333819514270305324688561049270188451486 21147118809388065383865436743357629844861804969660154563123016123042313614003 62930246401192497775895803344289731467512848459607878351483474766372717072170 72387197272893442098015714115198644477427427325214673723807387888482089247653 77713292922674286121689642433842212615229467646834807727260832188051558076507 09981257282535659403140674107145053077526002195201874023560057448303561661798 61884410043177391932703549065128308458939699325358798388911019573385219675317 44304617123012875316218376972594091512377127286897587775815981743533912623838 56891318749407882925680202903687650146268327404146267989090593999139072008668 13152327959295675141732681913571442322026768520084206924322401284323051403931 18979626641294646900049725069715540043368443224646343909861329258844132645237 00018024140684898981964083443341171876770172664139574056850248865093599257501 52483471350428752707577072001175666443593196701391561839123177816910486810554 86641171033353320729383984411167679719685156850021899792923469802870824357951 91771977135678534646580603614690229469306152174199854050309718399182634706016 16755110706693758601025701391444170431983397789955190084099233749081100187002 32092470774753064256464083708602000228504033289282060520062030789050191507276 39346252346659300692770630426425612810233260919192116983870103183654164035241 86276734604974270732789976436486933032848696626657420385705659400865290003030 23110278498632320836527984228015516860892875165241714612843846337970412777633 93926759186432 In both cases the expansion ends with our ninedigital !! Here also I detected a pattern as for all exponents +100 (starting with 1) it occurs. 1, 101, 201, 301, ...

826315794 23 = 12426797981687139976318194833800965760178941008925651967284591535884639168127 69048144289039195619651222534322636314920864225698824936480144264484392524652 8635188413679903582631579460068981211460106226499584

873264591 83 = 13034837372853334164238093382491785728595445636980606552677673905144344811209 36662942425790599835613708911373168631742552867591811425692124430330075409429 94755619948421750835314736209456882857274065496962124999162871439340633338995 04647924034305980271186771316971847687817508159884016625445308567446163138679 20006235542264658107219655488515060394364853003282248342845474873264591141569 44635942228225266830138874923545164149517423839878166331320624862020348714225 05181805029396362821890131393393223912184424551405029731746354590346537906403 67581873950902768490320199664827709531782965695130366107984725738284643788944 02735317759361314929421354339292024390717560432540358188871423020947904182402 43462672111318591628952881312605495336284911804271

With 873264591 I found the largest ninedigital with p < 119 !

[ March 1, 2015 ]
Nine- & pandigitals equal to the sum of two squares
Some statistics and curios

This topic is a continuation from wonplate193

Statistics and curios for the ninedigital variation

In total there are 65795 ninedigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 18% of the 9! possible ninedigitals. Here is the distribution list :

(*_1) 28763 (*_2) 25968 (*_3) 1464 (*_4) 7513 (*_5) 55 (*_6) 848 (*_7) 0 (*_8) 901 (*_9) 27 (*_10) 34 (*_11) 0 (*_12) 161 (*_13) 0 (*_14) 0 (*_15) 2           469738125 & 879463125 (*_16) 44

(*_17) 0 (*_18) 4           346978125, 647193825, 783961425 & 968417325 (*_19) 0 (*_20) 2           763498125 & 796843125 (*_21) 0 (*_22) 0 (*_23) 0 (*_24) 8 (*_25) 0 (*_26) 0 (*_27) 0 (*_28) 0 (*_29) 0 (*_30) 0 (*_31) 0 (*_32) 1           439817625

In total there are exactly 100 ninedigitals expressible as a sum of two squares whereby
the concatenation of its basenumbers forms a ninedigital (77 in total) or a pandigital (23 in total).

As a coincidence there are also 23 ninedigitals whereby the basenumbers A and B are the same.
They show up on the right side of the table.

One ninedigital stands out from the rest namely 317928645 because it is the
only one that has more than one solution. Beware this is a unique case!
This couple are the first two from an eightfold (*_8) solution for this ninedigital.
Note: this curio was first observed by Peter Kogel already in 2005.

Here is the complete list of all the hundred ninedigital solutions.
What is under construction is the list regarding the pandigitals. B.S. Rangaswamy sent me already one example
to wet your appetite. See at the bottom of the table.

1	175236849  =  34952 + 127682		612859437  =  137492 + 205892		143752968 = 2 * 84782
2	179684325  =  36542 + 128972		618542937  =  136592 + 207842		145897362 = 2 * 85412
3	189237465  =  45362 + 129872		689537412  =  109742 + 238562		162973458 = 2 * 90272
4	197485632  =  53762 + 129842		756928314  =  149672 + 230852		164275938 = 2 * 90632
5	218367945  =  54692 + 137282		783691245  =  185732 + 209462		178945362 = 2 * 94592
6	231649785  =  65282 + 137492		786425193  =  159482 + 230672		183974562 = 2 * 95912
7	234169785  =  76592 + 132482		814593672  =  158942 + 237062		219367458 = 2 * 104732
8	234718965  =  87932 + 125462		817439625  =  137492 + 250682		346581792 = 2 * 131642
9	234971685  =  56792 + 142382		824691537  =  103592 + 267842		423579618 = 2 * 145532
10	237916845  =  89732 + 125462		841769325  =  170582 + 234692		453968712 = 2 * 150662
11	238176549  =  69452 + 137822		867154293  =  169532 + 240782		461593728 = 2 * 151922
12	238971465  =  76592 + 134282		874932516  =  174962 + 238502		497638152 = 2 * 157742
13	243956817  =  27842 + 153692		891746325  =  137852 + 264902		537198642 = 2 * 163892
14	249813657  =  82592 + 134762		912645873  =  164972 + 253082		571963842 = 2 * 169112
15	251649873  =  53672 + 149282		916782345  =  185072 + 239642		618534792 = 2 * 175862
16	256847193  =  49682 + 152372		921847653  =  160982 + 257432		637459218 = 2 * 178532
17	268371954  =  69752 + 148232		925781634  =  143972 + 268052		639174258 = 2 * 178772
18	283974165  =  43592 + 162782		927814653  =  157982 + 260432		654279138 = 2 * 180872
19	286753194  =  82952 + 147632		934687125  =  143702 + 269852		654713298 = 2 * 180932
20	312897645  =  89342 + 152672		943725186  =  187052 + 243692		765421938 = 2 * 195632
21	317928645  =  25382 + 176492 =  29432 + 175862		948571236  =  136802 + 275942		913524768 = 2 * 213722
22	326785149  =  49652 + 173822		972354861  =  174692 + 258302		943256178 = 2 * 217172
23	328459617  =  54362 + 172892		973416285  =  174062 + 258932		958431762 = 2 * 218912
24	341978265  =  64592 + 173282
25	345172689  =  87452 + 163922
26	346297185  =  79532 + 168242
27	362794185  =  54962 + 182372
28	365984721  =  24362 + 189752
29	368529417  =  53762 + 184292
30	369471825  =  59642 + 182732
31	374921865  =  83522 + 174692
32	378129645  =  83492 + 175622
33	385267914  =  23672 + 194852
34	413629578  =  64532 + 192872
35	435869712  =  78242 + 193562
36	436521978  =  96272 + 185432
37	478192653  =  46982 + 213572
38	481379265  =  48572 + 213962
39	489731265  =  56972 + 213842
40	497163825  =  48962 + 217532
41	497231865  =  59762 + 214832
42	514926873  =  57632 + 219482
43	523687194  =  63452 + 219872
44	523861794  =  78632 + 214952
45	532978641  =  87962 + 213452
46	592876413  =  61982 + 235472
47	598314762  =  64712 + 235892
48	613259874  =  18752 + 246932
49	613478925  =  74582 + 236192
50	614829357  =  78692 + 235142
51	619423785  =  15962 + 248372
52	631548297  =  87962 + 235412
53	635127849  =  91682 + 234752
54	639218457  =  73562 + 241892
55	648912537  =  38642 + 251792
56	651429378  =  68132 + 245972
57	674921853  =  41972 + 256382
58	682143597  =  61892 + 253742
59	694725138  =  96872 + 245132
60	712839546  =  97352 + 248612
61	726389145  =  89762 + 254132
62	726439185  =  71642 + 259832
63	729563481  =  39842 + 267152
64	758394621  =  43952 + 271862
65	789631245  =  97412 + 263582
66	812596473  =  56132 + 279482
67	817453962  =  65492 + 278312
68	823415697  =  35162 + 284792
69	825697314  =  39152 + 284672
70	839147625  =  91562 + 274832
71	846352197  =  73592 + 281462
72	865972341  =  38462 + 291752
73	872156394  =  16352 + 294872
74	914367285  =  64712 + 295382
75	931427586  =  68312 + 297452
76	934816725  =  74312 + 296582
77	951246738  =  85172 + 296432

Another ninedigital that stands out from the rest is 934167285 because it is the only
one that can be written as a sum of two squares in two different ways such that both
their basenumbers forms a ninedigital when multiplied together.
A nice unique case!

In total there are 219 ninedigitals expressible in this way.
The smallest is 248635917 = 10614² + 11661² and 10614 * 11661 = 123769854
The largest is   987431562 = 21519² + 22899² and 21519 * 22899 = 492763581

Statistics and curios for the pandigital variation

In total there are 568801 pandigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 17,416 % of the 9*9! possible pandigitals. Here is the distribution list :

(*_1)  (23112)1 + (23193)2 + (23072)3 + (22132)4 + (24904)5 + (21085)6 + (22492)7 + (21296)8 + (20600)9 ( 201886 )tot (*_2)  (26329)1 + (26858)2 + (26881)3 + (25208)4 + (25865)5 + (24452)6 + (26628)7 + (24705)8 + (24742)9 ( 231668 )tot (*_3)  (1840)1 + (919)2 + (1592)3 + (1372)4 + (511)5 + (1434)6 + (1526)7 + (1324)8 + (1471)9 ( 11989 )tot (*_4)  (10254)1 + (10559)2 + (11129)3 + (10170)4 + (8929)5 + (9983)6 + (11204)7 + (10259)8 + (10737)9 ( 93224 )tot (*_5)  (60)1 + (0)2 + (110)3 + (116)4 + (5)5 + (57)6 + (105)7 + (96)8 + (107)9 ( 656 )tot (*_6)  (1408)1 + (740)2 + (1164)3 + (1127)4 + (377)5 + (1185)6 + (1257)7 + (1081)8 + (1156)9 ( 9495 )tot (*_7)  (5)1 + (0)2 + (7)3 + (7)4 + (0)5 + (1)6 + (5)7 + (5)8 + (4)9 ( 34 )tot (*_8)  (1601)1 + (1606)2 + (1864)3 + (1748)4 + (1197)5 + (1739)6 + (2088)7 + (1915)8 + (1992)9 ( 15750 )tot (*_9)  (32)1 + (19)2 + (25)3 + (17)4 + (5)5 + (23)6 + (30)7 + (29)8 + (25)9 ( 205 )tot (*_10)  (30)1 + (2)2 + (42)3 + (90)4 + (0)5 + (40)6 + (82)7 + (53)8 + (72)9 ( 411 )tot (*_11)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_12)  (251)1 + (133)2 + (281)3 + (266)4 + (73)5 + (270)6 + (297)7 + (290)8 + (294)9 ( 2155 )tot (*_13)  (1)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 2 )tot (*_14)  (0)1 + (0)2 + (1)3 + (4)4 + (0)5 + (4)6 + (3)7 + (1)8 + (7)9 ( 20 )tot (*_15)  (2)1 + (0)2 + (4)3 + (6)4 + (0)5 + (0)6 + (5)7 + (1)8 + (2)9 ( 20 )tot (*_16)  (88)1 + (71)2 + (111)3 + (114)4 + (49)5 + (128)6 + (142)7 + (156)8 + (153)9 ( 1012 )tot (*_17)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_18)  (9)1 + (3)2 + (11)3 + (14)4 + (2)5 + (7)6 + (9)7 + (14)8 + (10)9 ( 79 )tot (*_19)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_20)  (1)1 + (0)2 + (4)3 + (6)4 + (0)5 + (7)6 + (17)7 + (3)8 + (8)9 ( 46 )tot (*_21)  (1)1 + (0)2 + (1)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 3 )tot (*_22)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_23)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_24)  (11)1 + (9)2 + (14)3 + (12)4 + (3)5 + (12)6 + (23)7 + (8)8 + (22)9 ( 114 )tot (*_25)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_26)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_27)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_28)  (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (1)8 + (1)9 ( 3 )tot (*_29)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_30)  (0)1 + (0)2 + (2)3 + (2)4 + (0)5 + (0)6 + (1)7 + (1)8 + (0)9 ( 6 )tot (*_31)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_32)  (2)1 + (0)2 + (3)3 + (1)4 + (0)5 + (3)6 + (2)7 + (2)8 + (3)9 ( 16 )tot (*_33)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_34)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_35)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_36)  (0)1 + (0)2 + (1)3 + (0)4 + (0)5 + (1)6 + (1)7 + (0)8 + (2)9 ( 5 )tot       3049186725, 6137904825, 7261938450, 9132768450 & 9702436185 (*_37)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_38)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_39)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_40)  (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 1 )tot       4398176250 (*_41)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_42)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_43)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_44)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_45)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_46)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_47)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_48)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 1 )tot       7316984025 (*_subt)  (65037)1 + (64112)2 + (66319)3 + (62415)4 + (61920)5 + (60431)6 + (65919)7 + (61240)8 + (61408)9 ( 568801 )total

1	1023458976  =  194762 + 253802		1	1024397685  =  57692 + 314822
2	1024537896  =  193862 + 254702		2	1063789245  =  78422 + 316592
3	1024736985  =  165392 + 274082		3	1073982645  =  89462 + 315272
4	1032768549  =  146702 + 285932		4	1074893265  =  15692 + 327482
5	1038249657  =  185042 + 263792		5	1236785940  =  51962 + 347822
6	1038259476  =  198602 + 253742		6	1237694085  =  56822 + 347192
7	1058267349  =  184502 + 267932		7	1258047369  =  67952 + 348122
8	1063275498  =  159032 + 284672		8	1270365489  =  81752 + 346922
9	1073982645  =  198542 + 260732		9	1294037685  =  92582 + 347612
10	1075348296  =  145862 + 293702		10	1324958760  =  71942 + 356822
11	°  =  ° + °		11	°  =  ° + °
12	°  =  ° + °		12	°  =  ° + °

One lovely pandigital already popped up on my screen 1073982645
It combines in an elegant way the two numberformats i.e. ninedigital and pandigital !
This couple are the middle two from a fourfold (*_4) solution for this pandigital.

Then three more pandigitals with double solutions popped up later on !
They are fully pandigital solutions throughout. Enjoy them!

Two pandigitals that stand out from the rest are 5921803476 and 8097452136 because
they are the only ones that can be written as a sum of two squares such that both their
basenumbers form a pandigital when concatenated as well as when multiplied together.
A nice couplet !

While tracing for long(er) chains of pandigitals I stumbled across the following unique and
unexpectedly double expression. A thing of beauty !
Note that A and B are in descending order here. I didn't find a case whereby A and B are ascending.

Can you find longer chains of pandigital expressions using perhaps other operations than concatenation ?

More subcategories

All the pandigitals (41) equal to 2 x A² are A =
22887, 23124, 24957, 25941, 26409, 26733, 27276, 29685, 31389, 35367,
39036, 39147, 39432, 39702, 40293, 41997, 42843, 43059, 44922, 45258,
45624, 46464, 49059, 50889, 53568, 54354, 57321, 59268, 59727, 60984,
61098, 61611, 61866, 62634, 65436, 68823, 68982, 69087, 69696, 69732,
69798.
Note that two of them (see underlined) are palindromic !

[ April 27, 2014 ]
Order out of chaos using the ninedigits

And here is a video explaining the mysterious math from above
which is in fact the Erdös-Szekeres theorem.
https://www.youtube.com/watch?v=LBPj8E1JKaQ&feature=em-uploademail

[ December 15, 2013 ]
Nearing the end of the year 2013

Who likes to do this exercise for the coming years 2014, 2015, ...
And what about the pandigital version of this topic ?

[ January 1, 2013 ]
The ninedigital primes version of WONplate 181

[ December 30, 2010 ]
From a posting to [SeqFan]
by Eric Angelini

Happy New Year! "

[ April 2008 ]
Fractions using the same digits as their decimal representation
A webpage by Christian Boyer

The above link came from a reply in the SeqFan mailing list where
Alexander R. Povolotsky's topic was about approximating Pi
using just nine- and pandigitals. Here is his *best* combination !

Alexander R. Povolotsky [ October 8, 2022 ] writes Since then the better approximations were found per Using each number (1-9 EXACTLY ONCE) can you make 2 distinct 9 digits numbers, so the quotient of the two numbers is as close to Pi as possible? 429751836 / 136794258 = 3.14159265369164852899... (pi + 1.01855e-10) 467895213 / 148935672 = 3.14159265350479621759... (pi - 8.49969e-11) Pi = 3.1415926535897932384626433832795028842 I conjecture that the higher the pandigital number base then the better and better Pi approximations can be found and that in the base where n=infinity the actual Pi number could be achieved... Someone with good computer programming skills perhaps could check whether my conjecture is true or not.

[ March 31, 2008 ]
Blending palindromes with nine- & pandigitals using multiplication by 9
by B.S. Rangaswamy

Pandigitals (Kmil = 1000 million) total = 559

Palindrome	Pandigital
P9		P9 * 9
Palindrome	Ninedigital
P8		P8 * 9     

Palindrome	Pandigital
P9		P9 * 9
Palindrome	Ninedigital
P8		P8 * 9     

Palindrome	Pandigital
P9		P9 * 9
Palindrome	Ninedigital
P8		P8 * 9     

Palindrome	Pandigital
P9		P9 * 9
Palindrome	Ninedigital
P8		P8 * 9     

1 Kmil  # 81
2 Kmil  # 46

134050431	1206453879
134060431	1206543879
136717631	1230458679
136727631	1230548679
137606731	1238460579
137626731	1238640579
137818731	1240368579
137848731	1240638579
138929831	1250368479
138959831	1250638479
139828931	1258460379
139848931	1258640379
143050341	1287453069
143060341	1287543069
145030541	1305274869
145080541	1305724869
145272541	1307452869
145282541	1307542869
146717641	1320458769
146727641	1320548769
148919841	1340278569
148969841	1340728569
152676251	1374086259
152898251	1376084259
154030451	1386274059
154080451	1386724059
156252651	1406273859
156303651	1406732859
156373651	1407362859
157818751	1420368759
157848751	1420638759
158707851	1428370659
158747851	1428730659
158919851	1430278659
158969851	1430728659
167030761	1503276849
167080761	1503726849
167363761	1506273849
167414761	1506732849
167484761	1507362849
168929861	1520368749
168959861	1520638749
169818961	1528370649
169858961	1528730649
174898471	1574086239
176030671	1584276039
176080671	1584726039
176252671	1586274039
178050871	1602457839
178060871	1602547839
178252871	1604275839
178363871	1605274839
182565281	1643087529
183676381	1653087429
187050781	1683457029
187060781	1683547029
187303781	1685734029
187494781	1687453029
189272981	1703456829
189282981	1703546829
189373981	1704365829
189484981	1705364829
189515981	1705643829
215030512	1935274608
215080512	1935724608
215272512	1937452608
215282512	1937542608
216252612	1946273508
216303612	1946732508
217030712	1953276408
217080712	1953726408
217414712	1956732408
217484712	1957362408
218050812	1962457308
218060812	1962547308
218363812	1965274308
219272912	1973456208
219282912	1973546208
219373912	1974365208
219484912	1975364208
219515912	1975643208

224050422	2016453798
224060422	2016543798
225717522	2031457698
225727522	2031547698
226404622	2037641598
226818622	2041367598
226848622	2041637598
227959722	2051637498
240545042	2164905378
240656042	2165904378
255717552	2301457968
255727552	2301547968
264010462	2376094158
264404462	2379640158
266818662	2401367958
266848662	2401637958
268121862	2413096758
268454862	2416093758
277929772	2501367948
277959772	2501637948
279232972	2513096748
279565972	2516093748
286010682	2574096138
286232682	2576094138
286606682	2579460138
286626682	2579640138
315010513	2835094617
315494513 [Obs]	2839450617
315595513	2840359617
315989513	2843905617
316595613	2849360517
317010713	2853096417
318232813	2864095317
318343813	2865094317
326020623	2934185607
326090623	2934815607
326131623	2935184607
326494623	2938451607
327020723	2943186507
327090723	2943816507
327353723	2946183507
327595723	2948361507
328464823	2956183407
329050923	2961458307
329060923	2961548307
329464923	2965184307

3 Kmil  # 74
4 Kmil  # 90

336020633	3024185697
336090633	3024815697
336131633	3025184697
336494633	3028451697
341050143	3069451287
341060143	3069541287
344717443	3102456987
344727443	3102546987
347121743	3124095687
347232743	3125094687
360121063	3241089567
360212063	3241908567
360989063	3248901567
361202163	3250819467
361232163	3251089467
361323163	3251908467
364505463	3280549167
364989463	3284905167
365101563	3285914067
365494563	3289451067
365717563	3291458067
365727563	3291548067
378101873	3402916857
378545873	3406912857
389212983	3502916847
389656983	3506912847
412050214	3708451926
412060214	3708541926
412676214	3714085926
412787214	3715084926
413272314	3719450826
413282314	3719540826
416454614	3748091526
416545614	3748910526
416575614	3749180526
417565714	3758091426
417656714	3758910426
417686714	3759180426
420545024	3784905216
420656024	3785904216
422050224	3798452016
422060224	3798542016
423050324	3807452916
423060324	3807542916
425010524	3825094716
425494524	3829450716
427454724	3847092516
427696724	3849270516
428010824	3852097416
428565824	3857092416
431050134	3879451206
431060134	3879541206
432505234	3892547106
432747234	3894725106
432808234	3895274106
432858234	3895724106
435030534	3915274806
435080534	3915724806
435272534	3917452806
435282534	3917542806
435717534	3921457806
435727534	3921547806
436020634	3924185706
436090634	3924815706
436131634	3925184706
436494634	3928451706
438020834	3942187506
438090834	3942817506
438575834	3947182506
438696834	3948271506
439030934	3951278406
439080934	3951728406
439131934	3952187406
439686934	3957182406

445919544	4013275896
446202644	4015823796
448020844	4032187596
448090844	4032817596
448575844	4037182596
448646844	4037821596
448696844	4038271596
451030154	4059271386
451080154	4059721386
452373254	4071359286
452393254	4071539286
455919554	4103275986
455969554	4103725986
459121954	4132097586
459676954	4137092586
462010264	4158092376
462202264	4159820376
467101764	4203915876
480212084	4321908756
480989084	4328901756
485676584	4371089256
485767584	4371908256
486454684	4378092156
486646684	4379820156
486989684	4382907156
487545784	4387912056
487919784	4391278056
487969784	4391728056
512030215	4608271935
512080215	4608721935
512303215	4610728935
513010315	4617092835
513252315	4619270835
513353315	4620179835
513545315	4621907835
514232415	4628091735
514323415	4628910735
514353415	4629180735
519010915	4671098235
519787915	4678091235
519878915	4678910235
521454125	4693087125
521898125	4697083125
523181325	4708631925
523404325	4710638925
523676325	4713086925
524010425	4716093825
524373425	4719360825
526454625	4738091625
526545625	4738910625
526575625	4739180625
529010925	4761098325
529787925	4768091325
529878925	4768910325
531545135	4783906215
531878135	4786903215
533151335	4798362015
533181335	4798632015
534030435	4806273915
534080435	4806723915
534151435	4807362915
534181435	4807632915
536232635	4826093715
536595635	4829360715
536696635	4830269715
536989635	4832906715
537454735	4837092615
537696735	4839270615
541030145	4869271305
541080145	4869721305
541252145	4871269305
541292145	4871629305
542151245	4879361205
542181245	4879631205
543626345	4892637105
543747345	4893726105
546252645	4916273805
546303645	4916732805
546373645	4917362805
546818645	4921367805
546848645	4921637805
547020745	4923186705
547090745	4923816705
547353745	4926183705
547595745	4928361705
548020845	4932187605
548090845	4932817605
548575845	4937182605
548646845	4937821605
548696845	4938271605

5 Kmil  # 64
6 Kmil  # 78

623020326	5607182934
623090326	5607812934
623141326	5608271934
623191326	5608721934
623414326	5610728934
623565326	5612087934
624121426	5617092834
624363426	5619270834
624464426	5620179834
624656426	5621907834
625343526	5628091734
625434526	5628910734
625464526	5629180734
630121036	5671089324
630212036	5671908324
630989036	5678901324
632565236	5693087124
633020336	5697183024
633090336	5697813024
634020436	5706183924
634090436	5706813924
634262436	5708361924
634292436	5708631924
634515436	5710638924
634787436	5713086924
635121536	5716093824
635484536	5719360824
637565736	5738091624
637656736	5738910624
637686736	5739180624
642656246	5783906214
642989246	5786903214
644262446	5798362014
644292446	5798632014
645141546	5806273914
645191546	5806723914
645262546	5807362914
645292546	5807632914
647010746	5823096714
647343746	5826093714
648010846	5832097614
648565846	5837092614
652141256	5869271304
652191256	5869721304
652363256	5871269304
653262356	5879361204
653292356	5879631204
654707456	5892367104
654737456	5892637104
654808456	5893276104
654858456	5893726104
657030756	5913276804
657080756	5913726804
657363756	5916273804
657414756	5916732804
657484756	5917362804
657929756	5921367804
657959756	5921637804
658131856	5923186704
658464856	5926183704
659030956	5931278604
659080956	5931728604
659131956	5932187604
659686956	5937182604

668202866	6013825794
668424866	6015823794
671030176	6039271584
671080176	6039721584
673101376	6057912384
673252376	6059271384
674595476	6071359284
682010286	6138092574
682202286	6139820574
684232486	6158092374
684424486	6159820374
689323986	6203915874
689545986	6205913874
712030217	6408271953
712080217	6408721953
712303217	6410728953
712454217	6412087953
713010317	6417092853
713252317	6419270853
713353317	6420179853
713545317	6421907853
714232417	6428091753
714323417	6428910753
714353417	6429180753
719010917	6471098253
719878917	6478910253
723020327	6507182943
723090327	6507812943
723141327	6508271943
723191327	6508721943
723414327	6510728943
723565327	6512087943
724121427	6517092843
724363427	6519270843
724464427	6520179843
724656427	6521907843
725343527	6528091743
725434527	6528910743
725464527	6529180743
745020547	6705184923
745090547	6705814923
745383547	6708451923
745393547	6708541923
745606547	6710458923
745616547	6710548923
746010647	6714095823
746121647	6715094823
749010947	6741098523
749787947	6748091523
749878947	6748910523
753878357	6784905213
753989357	6785904213
755383557	6798452013
755393557	6798542013
756030657	6804275913
756080657	6804725913
756141657	6805274913
756191657	6805724913
756383657	6807452913
756393657	6807542913
758232857	6824095713
758343857	6825094713
761030167	6849271503
761080167	6849721503
762141267	6859271403
762191267	6859721403
764383467	6879451203
764393467	6879541203
765828567	6892457103
765838567	6892547103
768050867	6912457803
768060867	6912547803
768252867	6914275803
768363867	6915274803
769050967	6921458703
769060967	6921548703
769353967	6924185703
769464967	6925184703

7 Kmil  # 69
8 Kmil  # 57

781050187	7029451683
781060187	7029541683
783474387	7051269483
785101587	7065914283
785494587	7069451283
812050218	7308451962
812060218	7308541962
812676218	7314085962
812787218	7315084962
816454618	7348091562
816575618	7349180562
817565718	7358091462
817656718	7358910462
821565128	7394086152
821787128	7396084152
823151328	7408361952
823181328	7408631952
824010428	7416093852
824373428	7419360852
826454628	7438091652
826545628	7438910652
826575628	7439180652
829010928	7461098352
829787928	7468091352
829878928	7468910352
834020438	7506183942
834090438	7506813942
834262438	7508361942
834292438	7508631942
834515438	7510638942
834787438	7513086942
835121538	7516093842
835484538	7519360842
837565738	7538091642
837656738	7538910642
837686738	7539180642
840121048	7561089432
840212048	7561908432
840989048	7568901432
843787348	7594086132
844020448	7596184032
844090448	7596814032
845020548	7605184932
845090548	7605814932
845383548	7608451932
845393548	7608541932
845606548	7610458932
845616548	7610548932
846010648	7614095832
846121648	7615094832
849010948	7641098532
849787948	7648091532
849878948	7648910532
867050768	7803456912
867060768	7803546912
867151768	7804365912
867181768	7804635912
867383768	7806453912
867393768	7806543912
871050178	7839451602
871060178	7839541602
871262178	7841359602
873262378	7859361402
873292378	7859631402
874383478	7869451302
874393478	7869541302
879272978	7913456802
879282978	7913546802
879515978	7915643802

915010519	8235094671
915595519	8240359671
915989519	8243905671
916595619	8249360571
917010719	8253096471
917343719	8256093471
918232819	8264095371
918343819	8265094371
923050329	8307452961
923060329	8307542961
925010529	8325094761
925494529	8329450761
927454729	8347092561
927696729	8349270561
928010829	8352097461
928565829	8357092461
934030439	8406273951
934080439	8406723951
934151439	8407362951
934181439	8407632951
936232639	8426093751
936595639	8429360751
936696639	8430269751
936989639	8432906751
937454739	8437092651
937696739	8439270651
945141549	8506273941
945191549	8506723941
945262549	8507362941
945292549	8507632941
947010749	8523096741
947343749	8526093741
948010849	8532097641
948565849	8537092641
951434159	8562907431
951989159	8567902431
955141559	8596274031
955191559	8596724031
956030659	8604275931
956080659	8604725931
956141659	8605274931
956383659	8607452931
956393659	8607542931
958232859	8624095731
958343859	8625094731
963878369	8674905321
963989369	8675904321
966383669	8697453021
966393669	8697543021
967050769	8703456921
967060769	8703546921
967151769	8704365921
967181769	8704635921
967262769	8705364921
967292769	8705634921
967383769	8706453921
967393769	8706543921

[ January 22, 2006 ]
Pandigital... throughout

A unique construction : start with multiplying these two
5-digit factors which taken together form a pandigital number :

equals

as you can see the result of the multiplication is pandigital as well.
And now let us take the square of this pandigital 3410856297²

... an order_2 pandigital emerges since all the digits from 0 to 9
occur exactly two times !

Discover more of these gems at wonplate 167 !

[ October 26, 2005 ]
From Palindromic Squares to Pandigitals

Did you know that when we power up 264 with two exponents and add them up
that we arrive at a pandigital number... in two different ways !

B.S. Rangaswamy [ August 11, 2006 ] was enthused by this presentation
and found the following solutions :

232322 *   8 = 4317806592 = 464642 * 2 232322 * 18 = 9715064832 = 696962 * 2 423242 *   4 = 7165283904

One can play along with other than our 264 number namely 2016.
I found the next equation of some interest.

Or the next one with two consecutive integers

[ October 2005 ]
Figure this out : 4-1-4

It is a four digit number multiplied by a one digit number to equal another four digit number
    and only the nine digits from 1 to 9 can be used once ?

Two solutions to this 9-digit problem can be found.

[ October 2006 ]
Figure this out : 4-1-5

It is a four digit number multiplied by a one digit number to equal a five digit number
    and only the ten digits from 0 to 9 can be used once ?

Thirteen solutions to this pandigital problem can be found.

[ September 4, 2005 ]
Generating Pandigitals from Palindromes through Fibonacci iteration
by B.S. Rangaswamy

" I got inspired by your presentation of the derivation of 68 ninedigit numbers (with all numerals from 1 to 9)
from palindromes through Fibonacci iteration. I have developed it further by arriving at a dozen 10 digit numbers
(with all numerals from 0 to 9) from 2 to 10 digit palindromes. Some of the 10 digit numbers arrived at
together with their mother palindromes are :

75257 9135748206 799997 2067193845 8055508 4913860257 42944924 2361970854

It is interesting to note that 6300036 leads to pandigital 1467908532, which matches identically
with each stage of iteration of 630036 to 146798532. I came across another curio as well :

630036   146798532 6300036 1467908532 9530359   123894675 95300359 1238904675

Following closely resembling palindromes lead to closely matching pandigitals :

592070295 2960351478 952070259 4760351298

This task began at my son's residence in Florida US and was completed at Bangalore India.
I was thrilled at the discovery of each of these pandigitals.

I am grateful to you for your encouragement and guidance in this venture.
B.S.Rangaswamy "

In total there are 117 palindromes that yield pandigital numbers
The smallest one is 75257 and the largest one is 4376006734 ¬

4. Scintillating Grid
one pair each of digits 0, 3, 4, 6 & 7

Palindrome Pandigital     0346707643 0347606743 0436707634 0437606734 3067447603 3076446703 3467007643 3476006743 4067337604 4076336704 4367007634 4376006734 0693415287 0695213487 0873415269 0875213469 6134895207 6152893407 6934015287 6952013487 8134675209 8152673409 8734015269 8752013469

B.S. Rangaswamy (email) - go to topic 1

B.S. Rangaswamy (email) - go to topic 2

[ TOP OF PAGE]

( © All rights reserved ) - Last modified : December 12, 2022.

---

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

---