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EMBED
Palindromic Products
of Two Pandigital Numbers
rood Two ninedigitals rood comments



Introduction

A continuation of the topic started in WONplate 115

How many palindromes can you find
that are the products of two pandigital numbers ?



The Table

#Multiplicand
sorted by
MultiplierPalindromic productDate
1102368795429016735482970408257528040792Oct 9 2001
2102374689554217683095550518471748150555Oct 29 2001
3102486539769207548317092842147412482907Nov 14 2003
4102495673842186507934323934555554393234Nov 15 2003
5102543879659708231466122713697963172216Nov 15 2003
6102564387980497156328256141565651416528Nov 16 2003
7102568493780935261748301407883887041038Nov 16 2003
8102574368980396714528246642253522466428
odd digits are in the middle !
Nov 16 2003
9102598637484197056328638503251523058368Nov 16 2003
10102689574364107385926583160169610613856Nov 16 2003
11102738956463580412796532185257525812356Nov 17 2003
12102785639420194675832075722667662275702Nov 17 2003
13102845937684526170938693173301033713968Nov 17 2003
14102845963776039824517820389031309830287Nov 17 2003
15102846539718792546031932748331338472391Nov 17 2003
16102953486790835264179351807161617081539Nov 18 2003
17102957368465084721396700951637361590076Nov 18 2003
18102963578428497150632934168603068614392Nov 18 2003
19102973584660427985316222486257526842226
odd digits are in the middle !
Nov 18 2003
20102974835678012965438033372289822733308Nov 19 2003
21103268549760143729586210955727275590126Nov 19 2003
22103284796558016297435992201473741022995Nov 19 2003
23103286594789136452079206600597950066029Nov 19 2003
24103456278927609154382856340375730436582Nov 20 2003
25103462978549160825735086325455545236805Nov 20 2003
26103482975679526804138229670331330769228Nov 20 2003
27103487659264918527036718266401046628176Nov 20 2003
28103527694847205169834887042415142407884Nov 20 2003
29103527986480143569278297102349432017928Nov 20 2003
30103568279442815930764434372279722734344Nov 21 2003
31103569482719835470262054349393939434502Nov 21 2003
32103597482639567281044099070709070709904Nov 21 2003
33103765948225460873912641971723271791462Nov 22 2003
34103782456924736519082567216725276127652Nov 22 2003
35103798542621365408972217698313138967122Nov 23 2003
36103827496552716408935473412763672143745Nov 23 2003
37103829475679805412638286154143414516828Nov 23 2003
38103845276917260384591792409417149042971Nov 23 2003
39103862497550841329675280507475747050825Nov 23 2003
40103867452961825309746421637447447361246Nov 23 2003
41103924678556708439215893406313136043985Nov 23 2003
42103926745820918753642174017991997104712Nov 23 2003
43103942586720914875362173946245426493712Nov 23 2003
44103965874243106975824481654425244561844Nov 24 2003
45103982756446207589314804792503052974084Nov 24 2003
46103987654226980543712805643449443465082Nov 24 2003
47104273869556793024815922028457548202295Nov 24 2003
48104369285762915734086566470225220746656Nov 25 2003
49104536879220789541362173273773773723712Nov 26 2003
50104587932626859413072809170483840719082Nov 26 2003
51104593768220785349162174017991997104712Nov 26 2003
52104632758926107539482731703883883071372Nov 26 2003
53104682579326315798042754805615165084572Nov 27 2003
54104695237885203976418920450571750540298Nov 27 2003
55104725693845102986734723441577751443274Nov 27 2003
56104735869282675319048659071401041709568Nov 27 2003
57104783269548106325795040738099908370405Nov 27 2003
58104863725913278564091392439705079342931Nov 28 2003
59104873956245907861324814539037309354184Nov 28 2003
60104956278391526708439606302681862036069Nov 29 2003
61104965278321698057342277542627262457722Nov 29 2003
62105234798619286340572029594165614959202Nov 29 2003
63105273648989652307419438025533355208349Nov 30 2003
64105274863979816504328402671629261762048Nov 30 2003
65105286743948036172955057572239322757505Nov 30 2003
66105286973464837510296826545221225456286Nov 30 2003
67105346897245812630974826218525258126284Nov 30 2003
68105372469825713908642709538061608359072Nov 30 2003
69105396487265742903186929071053501709296Dec 1 2003
70105428379621940853672313188649468813132Dec 1 2003
71105462798357314906826044590457540954406Dec 1 2003
72105472869334682075913658018059508108563Dec 1 2003
73105489236779360258148371653055503561738Dec 1 2003
74105648372936490581273855170537350715583Dec 2 2003
75105648397280276519348481085601065801848Dec 2 2003
76105673492880293745168484920501050294848Dec 2 2003
77105678432923516704982485208529258025842Dec 2 2003
78105693784276093851248042647091907462408Dec 2 2003
79105792346823805697142518460567650648152Dec 3 2003
80105826437928409316573006456775776546003Dec 3 2003
81105829476332789054613470048477748400743Dec 3 2003
82105836742978169540328273209541459023728Dec 3 2003
83105839674276019358248045864109014685408Dec 3 2003
84105847236914520368791536940915190496351Dec 3 2003
85105869342778532160948314148259528414138Dec 3 2003
86105893746264798201536861724307034271686Dec 3 2003
87105896342712678904531342649619169462431Dec 3 2003
88105897326478143690528275207901097025728Dec 3 2003
89105923486774059328617844622309032264487Dec 4 2003
90105937246883097452168803115297925113088Dec 4 2003
91105947638238056149724031959181819591304Dec 4 2003
92105967283427961540382963008473748003692Dec 4 2003
93105974628340317596284272642279722462724Dec 4 2003
94105976482374562810397901904355534091097Dec 4 2003
95106253489727840159362958114085804118592Dec 5 2003
96106274538959638710246338076431346708336Dec 5 2003
97106295437858903624176261186519156811626Dec 5 2003
98106295473820389615742167323865683237612Dec 5 2003
99106352798440293857614285364515154635824Dec 6 2003
100106357298439720514864224566651566654224Dec 6 2003
101106385247919420756382066081981891806602Dec 6 2003
102106389247553960724815740840907090480475Dec 6 2003
103106435728979508146328462507507057052648Dec 7 2003
104106453297864937510826912812177712182196Dec 7 2003
105106493285731705428693376415275725146733Dec 7 2003
106106524938737684015924014287485847824104Dec 7 2003
107106537298463051724896717360429240637176Dec 7 2003
108106537984260758249136473061385831603746Dec 7 2003
109106539427875243019868016348281828436108Dec 7 2003
110106572349825104378692675432627262345762Dec 7 2003
111106582439730841672593287180709070817823Dec 8 2003
112106593874223086794512460910869680190642Dec 8 2003
113106725349821769053842323309885889033232Dec 8 2003
114106745892358194073626211978315138791126Dec 8 2003
115106752493879380152468474029233329204748Dec 8 2003
116106759832423601579482519700669660079152Dec 8 2003
117106794835243862019574684237151517324864Dec 8 2003
118106827354930149856273220829395939280223Dec 8 2003
119106847925348791306255213249845489423125Dec 9 2003
120106853429734675812093705229449449225073Dec 9 2003
121106873542927615390482951354619164531592Dec 9 2003
122106874925372156843097711757215127571177Dec 9 2003
123106957423845826970134901534665664351094Dec 10 2003
124107234659882159347068810329631369230188Dec 10 2003
125107236845940153789264305965711175695034Dec 10 2003
126107284965379523104868531633545453361358Dec 11 2003
127107329564841093765284410575943495750144Dec 11 2003
128107352894624531086972633483193913843362Dec 11 2003
129107354689224059876312582940543450492852Dec 11 2003
130107362458985246903719152317195917132519Dec 11 2003
131107368924551248607395502507857587052055Dec 11 2003
132107496352856813029476107193459543917016Dec 12 2003
133107523946825178369042707277613167727072Dec 12 2003
134107526983425018746932690190385830910962Dec 12 2003
135107532846934802965173742461925291642473Dec 12 2003
136107542368928930456173111249789879421113Dec 12 2003
137107568329475039614828071886005006881708Dec 13 2003
138107586439223760549812556312947492136552Dec 13 2003
139107632584970215864937557515043405157557Dec 13 2003
140107639582457241683096161470863680741616Dec 13 2003
141107649235823867510942569319313139139652Dec 13 2003
142107649523846089572134961520491940251694Dec 13 2003
143107659238425716394082768607401047068672Dec 14 2003
144107683524959270381646382443617163442836Dec 14 2003
145107684923552430786195646005199915006465Dec 14 2003
146107689234546573812095015498171718945105Dec 14 2003
147107694582340692137584382322759572232834Dec 14 2003
148107843265925317649082730337961697330372Dec 14 2003
149107894263546752819035044360975790634405Dec 15 2003
150107896524337605149284057464903094647504Dec 15 2003
151107945682347639821055142512987892152415Dec 15 2003
152107956823482465301978902692041402962098Dec 15 2003
153107983546240953178624422269455549622244Dec 15 2003
154108254639748096713255206692363632966025Dec 16 2003
155108259643781739650248849105411145019488Dec 16 2003
156108264793552164708395647601381831067465Dec 16 2003
157108267954374208193568034369309039634308Dec 16 2003
158108276345974395860128055311883881135508Dec 16 2003
159108357264918402763591994073129213704991Dec 17 2003
160108523497656970238416182609531359062816Dec 18 2003
161108523764935029814673801567371737651083Dec 18 2003
162108542379676054932188255183319133815528Dec 18 2003
163108564392741587369024514907461647094154Dec 18 2003
164108567243968194250737403661851581663047Dec 18 2003
165108574369241089236574461237941497321644Dec 18 2003
166108592673424568071932667912611162197662Dec 19 2003
167108594726357468193026240742691962470426Dec 19 2003
168108647325979083451628592205541455022958Dec 19 2003
169108652793425916840732815937141417395182Dec 19 2003
170108659273424187560932628202795972028262Dec 19 2003
171108724359640851672394441571919191751444
odd digits in the middle !
Dec 20 2003
172108764593225489063712772307645467032772Dec 22 2003
173108923456731648750293447291281821927443Dec 22 2003
174108945736227085394162950838207028380592Dec 22 2003
175109268374548231796055270209953599020725Dec 23 2003
176109386527427154639082970351671761530792Dec 25 2003
177109458376221047598362303835939395383032Dec 25 2003
178109463275860483951276620771439341770266Dec 25 2003
179109465837219328075462115763961693675112Dec 25 2003
180109473652880952364718862151067601512688Dec 25 2003
181109537824679520186438710468233328640178Dec 26 2003
182109538746236951870424047661555551667404Dec 26 2003
183109546728329603475813242963921293692423Dec 26 2003
184109625483719850243762176092573752906712Dec 26 2003
185109628347547610289355219437345437349125Dec 26 2003
186109635274826793518042937514713174157392Dec 26 2003
187109643752874918503268214345851585434128Dec 26 2003
188109738642552418306795752313829283132575Dec 27 2003
189109758236423059714682530993615163990352Dec 27 2003
190109783564243861079524815225639365225184Dec 28 2003
191109837564261340957286737541333331457376Dec 29 2003
192109843276547056238915168811461641188615Dec 29 2003
193109847653261024758936703426555556243076Dec 29 2003
194109854237678591062438633561245421653368Dec 29 2003
195109862457372835140968001847563657481008Jan 2009
196109875346236504718924010968629268690104Jan 2009
197109875436223967845012633477425247743362Jan 2009
 
198120346897543908615275284265621265624825Nov 20 2005
199120376954870423896518477414207024147748Nov 20 2005
200120378459652370986416304338671768334036Nov 20 2005
201120378496549261587035930035781875300395Nov 20 2005
202120435976868354079218232290297920922328Nov 20 2005
203120456987312590364871516597421247956151Nov 20 2005
204120486359770895263148541912175712191458Nov 21 2005
205120534976868507943218257603345433067528Nov 21 2005
206120537984669514287038379112059502119738Nov 21 2005
207120548736940859173264925521727271255294Nov 27 2005
208120564389753476209186447326523256237446Nov 27 2005
209120589764320135786942428169801089618242Nov 27 2005
210120675948330497528613680318185818130863Nov 27 2005
211120697845374820691539030696251526960309Nov 28 2005
212120745689342106973855084235581855324805Nov 28 2005
213120746389543169280755212534787874352125Nov 28 2005
214120754389656240897316791335225225331976Dec 4 2005
215120768359451028379646162613691963162616Dec 4 2005
216120785643953269078146434139903099314346Dec 5 2005
217120836475925849361073123545695965453213Dec 5 2005
218120836579451432806796214964441444694126Dec 5 2005
219120854697373845910268924625131315264298Dec 5 2005
220120857496371820594368680057217127500868Dec 5 2005
221120857694332647098513945653051503565493Dec 5 2005
222120879356438410597264643048275728403464Dec 5 2005
223120894657328670345913466091643461906643Dec 6 2005
224120895476329456780313561191485841911653Dec 6 2005
225120897564321705968342624198703078914262Dec 6 2005
226120936458765947028317975400065600045797Dec 6 2005
227120938765450124376896061980257520891606Dec 6 2005
228120947685351936780426281633373733361826Dec 6 2005
229120947865375918046239182125629265212819Dec 6 2005
230120954378656983472016892400447440042986Dec 6 2005
231120968543780452913679732271803081722379Dec 6 2005
232120975483669875021438453164509054613548Dec 6 2005
233120983746542830765915181826525256281815Dec 6 2005
234120983746548975210635925204467644025295Dec 6 2005
235123045986750162973486172352567652532716Dec 6 2005
236123047985635907186244418306935396038144Dec 6 2005
237123054968770329158648654352415142534568Dec 6 2005
238123067958437428015964606189511159816064Dec 7 2005
239123069485725048163793082664635364662803Dec 7 2005
240123069754850348219676196343049403436916Dec 7 2005
241123074956853489021676583159031309513856Dec 7 2005
242123076459865902837418111087919197801118Dec 7 2005
243123475908668927401538510873531353780158Dec 7 2005
244123490586730256847193736435811185346373Dec 8 2005
245123579480650249863716209852057502589026Dec 8 2005
246123987560467984305128429208137318029248Dec 11 2005
247124305986756780439217058148521258418507Dec 12 2005
248124359780648269357016002766647466672006Dec 12 2005
249124365907818235076492267821841481287622Dec 12 2005
250124385096737560814924672005595955002764Jan 13 2006
251124506378917493256082178021969691208712Jan 13 2006
252124586073968195047328496153205023516948Jan 13 2006
253124589603725843106793219782433342879123Jan 13 2006
254124630759848502613976044917631367194406Jan 14 2006
255124695803721748903562711997009007991172Jan 14 2006
256124850963762490738517802028925298202087Jan 15 2006
257124907356870492835168805073713173705088Jan 16 2006
258124907836546378209155793001765671003975Jan 16 2006
259124975086357864301297231596047406951327Jan 16 2006
260125068473917365942082171931873781391712Jan 17 2006
261125086493727654810393459243265623429543Jan 17 2006
262125098674327968354013498804009004088943Jan 17 2006
263125347906848926051376132778127218772316Jan 17 2006
264125386470932478910564072415973795142704Jan 17 2006
265125394068742316798055306275481845726035Jan 18 2006
266125670983434572869014344806447446084434Jan 19 2006
267125673904821856407392746780061600876472Jan 19 2006
268125697043867908523418535900641460095358Jan 19 2006
269125730846962845790137901654417144561097Jan 19 2006
270125749806334690725184362351971791532634Jan 19 2006
271125804796313269058471669311197911139661Jan 20 2006
272125807364960752194837643073543453703467Jan 20 2006
273125843067958694712037386322631362236837Jan 20 2006
274125863049741793806255260295913195920625Jan 20 2006
275125870493632514076894092562907092652904Jan 20 2006
276125873406913740268591729534419144359271Jan 20 2006
277125940783648920173566161044991994401616Jan 21 2006
278125970634867283490518475744011104475748Jan 21 2006
279125973860437205498164686920231320296864Jan 21 2006
280125983704653718249066767624021204267676Jan 21 2006
281126034597853401768926730470467640740376Jan 21 2006
282126094357817268953042177517543457157712Jan 22 2006
283126095348739820647155021198387838911205Jan 22 2006
284126308745959708461237541700857580071457Jan 22 2006
285126354078925684391073245327575757235423Jan 22 2006
286126375940820367845192574005597955004752Jan 22 2006
287126394850728356104793584065631365604853Jan 22 2006
288126408537920619435782606472729272746062Jan 22 2006
289126450789313695028471731747159517471371Jan 23 2006
290126534980734510798624366823237323286634Jan 23 2006
291126548370940862791355171119675769111715Jan 23 2006
292126735809435291804764472735441445372744Jan 24 2006
293126750489374168250939400862095902680049Jan 24 2006
294126780349541976028535321735567655371235Jan 24 2006
295126780394541672093855283204497944023825Jan 24 2006
296126793580451724630896558351145411538556Jan 24 2006
297126835049726103547893310844793974480133Jan 24 2006
298126875430921356794082709652451542569072Jan 25 2006
299126905783447862513096074029719179204706Jan 25 2006
300126943057821860935742775094029204905772Jan 25 2006
301127038964542593687015411057891987501145Jan 26 2006
302127045683974682035919488030327230308849Jan 26 2006
303127056483941908723655324775071705774235Jan 26 2006
304127083946542971086535460935261625390645Jan 26 2006
305127304958631879025644058358039308538504Jan 26 2006
306127490638540623718595179143821283419715Jan 28 2006
307127590843664590821738241197433347911428Jan 28 2006
308127634509865432907818351497113117941538Jan 28 2006
309127658093467439158028609154333334519068Jan 29 2006
310127680543957614209837356213647463126537Jan 29 2006
311127694530854620198376974700603060074796Jan 29 2006
312127804369546723095815971415801085141795Jan 29 2006
313127843605921749536082780539119119350872Jan 29 2006
314127850369465870431928421559053509551248Jan 29 2006
315127935806432875096414205901969691095024Jan 30 2006
316127946305820187563942582924229224292852Jan 30 2006
317128049376543502189675570428263628240755Jan 31 2006
318128063974541629705835331265585855621335Jan 31 2006
319128064537921730549685570428263628240755Jan 31 2006
320128065973421790346582790601945491060972Jan 31 2006
321128079435616534789022117766445446677112Jan 31 2006
322128079536421706543982780164089804610872Jan 31 2006
323128093475632861975044209404597954049024Jan 31 2006
324128345760931940782564099464041404649904Jan 31 2006
325128470536956940128737315128909098215137Feb 1 2006
326128475630950178326946446692211122966446Feb 1 2006
327128537640940162987355162455645465542615Feb 1 2006
328128540369739021784655015874625264785105Feb 1 2006
329128574093648073219566180970631360790816Feb 2 2006
330128593064721490835762763572433342753672Feb 2 2006
331128594076319836725042550885333335880552Feb 2 2006
332128630579453712096846909018137318109096Feb 2 2006
333128635709417092536482198710555550178912Feb 2 2006
334128639740539457028615075741921291475705Feb 2 2006
335128695730420197456382599326401046239952Feb 3 2006
336128750643940918263755268252805082528625Feb 3 2006
337128930745649607258316395900801080095936Feb 3 2006
338128936057435097468214525329175719235254Feb 3 2006
339129358740665812074938513367129217633158Feb 5 2006
340129367580421057349682724138377738314272Feb 5 2006
341129437056850864137926583704309034073856Feb 5 2006
342129468307540762915835277505721275057725Feb 5 2006
343129470853650831297466581171471741711856Feb 6 2006
344129560873453107864296880701281821070886Feb 6 2006
345129670835435840297164647441273721447464Feb 7 2006
346129735840647081259366108126759576218016Feb 7 2006
347129765380451863207946730048907098400376Feb 7 2006
348129837405616480739522139816461646189312Feb 8 2006
349129843570665897102438556315071705136558Feb 8 2006
350129845370618059346722344922567652294432Feb 8 2006
351129860374542703681595545516083806155455Feb 8 2006
352129874065375960824139865341033301435689Feb 8 2006
 
#To avoid pagesize overhead this list is
continued without HTML formatting
See text file propan3.txt


Contributions

[ October 7, 2008 ]
Peter Kogel (email) presents some nice results.

The following three examples are particularly noteworthy :
Each palindrome is pandigital as well !

[7105364289]824635017 = 1708463529 x 4158920673 (815)

[8251306479]746031528 = 1367208459 x 6035148792 (446)

[8754126039]306214578 = 2384507961 x 3671250498 (1155)

There are also similar examples where the pandigital palindrome is the product
of a ninedigital number and a pandigital number (3 examples) eg.:

[5132704689]864072315 = 631792485 x 8124035679

[6108425739]375248016 = 658731294 x 9273015864

[6548703219]123078456 = 862453719 x 7593106824






Observations by PDG

Two pandigitals in the list can be multiplied
with two other pandigitals to arrive at a palindrome.

1728603594 *
1367049528 = 2363086727276803632 palindrome (444)
       x
4790813562 = 8281417541457141828 palindrome (828)

2170654398 *
1280795364 = 2780164089804610872 palindrome (322)
       x
3908145276 = 8483232731372323848 palindrome (1117)


Three palindromes can be expressed as products of
two pandigitals in two different but math-related ways.

2174017991997104712 =
1039267458 * 2091875364 N1 * N2 (42)
        x                x
1045937682 * 2078534916 N2/2 * N1*2 (51)

6495526015106255946 =
1308652947 * 4963520718 N1 * N2 (378)
        x                x
2481760359 * 2617305894 N2/2 * N1*2 (1187)

8272731805081372728 =
2015793846 * 4103957268 N1 * N2 (1007)
        x                x
2051978634 * 4031587692 N2/2 * N1*2 (1041)


Two palindromes are connected through a reversal
transformation of one of its pandigital factors
(Resp. index numbers 87 and 1226). This is a unique case !

1342649619169462431
=
1058963427
*
1267890453

[ REVERSE DIGITS ! ]

3540987621
*
2603581749
=
9219250743470529129


There are six cases whereby the sum of the pandigital factors is pandigital as well !

3220829395939280223 =
1068273549 * 3014985627 (118)

1068273549 + 3014985627 = 4083259176

2115763961693675112 =
1094658372 * 1932807546 (179)

1094658372 + 1932807546 = 3027465918

2893714315134173982 =
1672034859 * 1730654298 (789)

1672034859 + 1730654298 = 3402689157

7247133287823317427 =
1892463507 * 3829470561 (940)

1892463507 + 3829470561 = 5721934068

9758213125213128579 =
2064378159 * 4726950381 (1048)

2064378159 + 4726950381 = 6791328540

5595037421247305955 =
2150698473 * 2601497835 (1103)

2150698473 + 2601497835 = 4752196308


There are two cases whereby the difference of the pandigital factors is pandigital as well !

4031959181819591304 =
3805614972 * 1059476382 (91)

3805614972 1059476382 = 2746138590

5438225545455228345 =
2907438165 * 1870452693 (934)

2907438165 1870452693 = 1036985472


This is a really beautiful gem !
Only the third and fourth digit of the pandigital factors need be swapped to arrive at the palindrome.

4270936083806390724 =
2057643918 (1044)
    2 ways
2075643918

The palindrome is a concatenation of five 3-digit numbers separated by zeroes :
4270936083806390724A | B | C | D | E

A + B + D = 2002

D - A = 212 = B - E

E - A = 297 = B - D

A lot of small palindromes pop up when the pandigitals are grouped into five sets of two digits.
20 | 57 | 64 | 39 | 18A | BN | C | D | E
20 | 75 | 64 | 39 | 18A | BR | C | D | E

Here they are sorted from smaller to larger :
A + BN = 77
A + D + E = 77

BN + C = 121
C + D + E = 121

A + BN + C = 141
A + C + D + E = 141

2A + (BN + BR) + 2C + 2D + 2E = 414

Some equalities can be detected :
BN = 57 = D + E

Playing around further (this may become politically incorrect...)

The largest factor of the palindrome is 115313551 which is a near-palindromic !
Break this down into three sets of 3-digit numbers
115 | 313 | 551
The middle part is palindromic prime 313 which when added
to the next palprime 353 gives 666, the numbers of the Beast !
Note that the sum of the left and right parts of the factor equals 666 as well !
115 + 551 = 666

The second pandigital gives another surprise when stripped of its last digit '8'.
207564391_[8]
Factorisation of this number gives us three primes
131 x 139 x 11399
Note the use of only three distinct digits 1, 3 and 9 as indicated
by the last three digits of the 'crippled' pandigital 207564391.

The second pandigital when stripped of its 'zero' digit is also in for a surprise !
2_[0]_75643918
Divide this ninedigital now into three sets of 3-digit numbers :
275 | 643 | 918
and you will immediately notice that
275 + 643 = 918







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