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Palindromic Products
of Two Ninedigital Numbers
rood Two pandigitals


Introduction

A continuation of the topic started in WONplate 97

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

On [ June 14, 2002 ] I (pdg) ended my search for these
extraordinary ninedigitals and found in total 220 solutions.
All the palindromic products are of length 17.
None of length 18 is discovered !
Can this phenomenon be explained mathematically ?

(answered by Jean Claude Rosa [ December 16, 2004 ]
see explanation after the table)


The Table

#Multiplicand
sorted by		MultiplierPalindromic productDate
112345678953872169466508850505880566Apr 09 2001
212358674957189264370678352525387607Sep 23 2001
312379586419246753823826685158662832Sep 26 2001
412438756913768245917125986368952171Sep 27 2001
512457936864573928180445791519754408Sep 27 2001
612463957823657981429487208180278492Sep 27 2001
712465387932814597640904668786640904Sep 27 2001
812478569327398465134189364546398143Sep 27 2001
912485937618975463223692644944629632Sep 27 2001
1012486379542968715353652368586325635Sep 27 2001
1112495673851892436764843096169034846Sep 27 2001
1212534768939182754649114677377641194Sep 27 2001
1312549873635784162944908672127680944Sep 27 2001
1412576943869758243187734550305543778Sep 27 2001
1512584369758764392173951283538215937Sep 28 2001
1612594673849516382762364268786246326Sep 28 2001
1712634975834925176844127876367872144Sep 28 2001
1812645798356137482970990328582309907Sep 28 2001
1912647358924963851731572679197627513Sep 28 2001
2012684753956824719372080757975708027Sep 28 2001
2112739546821869375427860593139506872Sep 28 2001
2212748536915682743919993203930239991Sep 29 2001
2312768495354876231970068690909686007Sep 29 2001
2412786459361274385978348244144284387Sep 29 2001
2512796358434795281644525289398252544Sep 29 2001
2612834957637821465948543691119634584Sep 29 2001
2712846593742731968554896023732069845Sep 29 2001
2812873956432864759142309947574990324Sep 29 2001
2912879645349715638264031978587913046Sep 29 2001
3012935478636187952446810848384801864Sep 30 2001
3112953468758932764176338371517383367Sep 30 2001
3212974538646389752160188562926588106Sep 30 2001
3312983465723481657930487329992378403Sep 30 2001
3413245896758927436178054673137645087Sep 30 2001
3513254798634657128945937326362373954Sep 30 2001
3613259864767281593489214482528441298Sep 30 2001
3713264589751429763868219471517491286Sep 30 2001
3813285467951286793468136904740963186Sep 30 2001
3913287564972934568196912280708221969Oct 1 2001
4013294586717584239623377519791577332Oct 1 2001
4113485726939467281553224497979442235Oct 1 2001
4213485769234215896746142768586724164Oct 1 2001
4313495268719783542626698422322489662Oct 2 2001
4413498257636127485948765811111856784Oct 2 2001
4513524896758237946178766220502266787Oct 2 2001
4613526894738129674551577609190677515Oct 2 2001
4713569247835761942848526266366262584Oct 2 2001
4813572964846938215763709074947090736Oct 2 2001
4913594687261975283484253459195435248Oct 3 2001
5013596248727165834936935344744353963Oct 3 2001
5113625948747196235864309348784390346Oct 3 2001
5213647298542673598158237933133973285Oct 3 2001
5313725489649365278167756261116265776Oct 4 2001
5413754296816893527423235858985853232Oct 4 2001
5513798465247682915365795104740159756Oct 5 2001
5613842967541863259757951174347115975Oct 5 2001
5713845692723861547933037965956973033Oct 5 2001
5813852694749527613868609091319090686Oct 6 2001
5913879625415643792821712998389921712Oct 6 2001
6013897624537826194552569424742496525Oct 6 2001
6113928564749126573868426266166262486Oct 6 2001
6213945728643897125661217739993771216Oct 6 2001
6313952648718743952626152778587725162Oct 6 2001
6413965742831927654844589341514398544Oct 8 2001
6513976524859314278682900748584700928Oct 8 2001
6613986724535816724950095866366859005Oct 8 2001
6714235768917952436825556674147665552Oct 8 2001
6814268539754892367178323391919332387Oct 8 2001
6914293785617865439225536475757463552Oct 9 2001
7014368257929473815642348738383784324Oct 9 2001
7114523879646592378167670208980207676Oct 11 2001
7214528736919654327828555255755255582Oct 11 2001
7314568923721853764931838583338583813Oct 11 2001
7414586273919465273828392581518529382Oct 12 2001
7514596732827943561840788470507488704Oct 12 2001
7614627958352613784976963225152236967Oct 12 2001
7714657328925189346736920853935802963Oct 13 2001
7814673589242165379861871746164717816Oct 13 2001
7914675839245736912867122757775722176Oct 13 2001
8014723956857482193684636533533563648Oct 13 2001
8114726839537285469154909711911790945Oct 13 2001
8214762953815437826922790792529709722Oct 14 2001
8314765328937824169555848630303684855Oct 14 2001
8414796523857693824185366804140866358Oct 14 2001
8514829735631548276946785260506258764Oct 14 2001
8614836925743295167864236718781763246Oct 14 2001
8714862759361527948391447508580574419Oct 14 2001
8814876923538916742557896140104169875Oct 15 2001
8914896523724137586935956613531665953Oct 15 2001
9014927385616853974225158577177585152Oct 15 2001
9114953628764132985795902085558020959Oct 16 2001
9214957326843758619265451196769115456Oct 16 2001
9314965237827341596840917349794371904Oct 16 2001
9414967258357843129686575306160357568Oct 16 2001
9515239467853142978680987071117078908Oct 18 2001
9615239678456893142786703319791330768Oct 18 2001
9715247398643185279665846317171364856Oct 18 2001
9815269734856342789186033944744933068Oct 19 2001
9915276893432674859149917033933071994Oct 19 2001
10015278493645178293669025626962652096Oct 19 2001
10115279648325649837139192048984029193Oct 19 2001
10215284793617963485227456816361865472Oct 20 2001
10315294637843215689766096832123869066Oct 20 2001
10415298346725834617939522694149622593Oct 20 2001
10515386942746583297171677452325477617Oct 20 2001
10615439278616547893225548753335784552Oct 21 2001
10715483762949671852376910718381701967Oct 21 2001
10815487296338174926559122639793622195Oct 21 2001
10915673892451382946780537077777073508Oct 22 2001
11015732894642781395667307518781570376Oct 23 2001
11115732964818672935429378063536087392Oct 23 2001
11215742683936748912557852651315625875Oct 24 2001
11315763982429741358646884225352248864Oct 24 2001
11415826374945869172372594271717249527Oct 25 2001
11515826974348125736976168480108486167Oct 25 2001
11615827469356982471390188831513888109Oct 25 2001
11715924678338617495261497118781179416Oct 27 2001
11815942873638794215661849127572194816Oct 27 2001
11915963827427931485644589341514398544Oct 27 2001
12016243875927598431644830549794503844Oct 27 2001
12116273548932846791553453386768335435Oct 27 2001
12216293784534892651756853334743335865Oct 28 2001
12316375982429547381648386740104768384Oct 28 2001
12416379584237659214861684227972248616Oct 28 2001
12516387452948521769379514820902841597Oct 28 2001
12616487529338761495263908128782180936Oct 28 2001
12716527389428934567147821285758212874Oct 29 2001
12816725834936712598461404885958840416Oct 29 2001
12916725938426538794144388623532688344Oct 29 2001
13016752983437425168962698323332389626Oct 29 2001
13116795324849158273682562917171926528Oct 29 2001
13216847325949376518283186229392268138Oct 29 2001
13316859732438742156965318239793281356Oct 29 2001
13416924753839148625766258085158085266Oct 30 2001
13517236985436751298463348159395184336Oct 30 2001
13617256834921349658736842753535724863Oct 30 2001
13717256834952387694190404578787540409Oct 30 2001
13817269835427931854648237853135873284Oct 30 2001
13917298354627368451947342918581924374Oct 30 2001
14017354298637529184665129267576292156Oct 30 2001
14117362945839148657267973601310637976Oct 30 2001
14217389426532168594755939341314393955Oct 31 2001
14317395246823416597840733749794733704Oct 31 2001
14417428369531862754955531586568513555Oct 31 2001
14517498562356179824398306615551660389Oct 31 2001
14617498632532764819557333953535933375Oct 31 2001
14717526389436718245964353827572835346Oct 31 2001
14817529864345162387979169053135096197Oct 31 2001
14917538624935821967462826804940862826Oct 31 2001
15017564832931768249555800399399300855Oct 31 2001
15117569428338145697267019809190891076Oct 31 2001
15217584923639428756169335166366153396Oct 31 2001
15317598432647591826383754154745145738Oct 31 2001
15417859246348591273686780352325308768Nov 4 2001
15517923845646279851382951290909215928Nov 4 2001
15617984326532146987557814191919141875Nov 4 2001
15717985263425734819646284750905748264Nov 4 2001
15818246937528574613952139919391993125Nov 5 2001
15918279346528716394552491692529619425Nov 6 2001
16018296754331496872557629053735092675Nov 6 2001
16118354269751863924795192446164429159Nov 6 2001
16218369527434862517964040797779704046Nov 7 2001
16318452367937561892469310585758501396Nov 8 2001
16418453762943689715280623964546932608Nov 8 2001
16518475269341762358977157082728075177Nov 9 2001
16618524973637594286169643315751334696Nov 9 2001
16718627945334516978264298038183089246Nov 10 2001
16818753946246572831987342438383424378Nov 11 2001
16918936754242815367981078409790487018Nov 11 2001
17019245738643927561884541837173814548Nov 12 2001
17119247685328137496554158167776185145Nov 12 2001
17219274638527984563153939233733293935Nov 13 2001
17319347658225643891749614925152941694Nov 14 2001
17419387246528169745354613379597331645Nov 15 2001
17519428756347269851391839442124493819Nov 15 2001
17619467532831689457261691554745519616Nov 16 2001
17719536428749362158796436029392063469Nov 16 2001
17819547236841382795680891930503919808Nov 17 2001
17919574328634725891667973601310637976Nov 17 2001
18019625483728973164556861236763216865Nov 17 2001
18119672834528136745955352954545925355Nov 18 2001
18219678435221456938742223897779832224Nov 18 2001
18319682345734672159868242943534924286Nov 18 2001
18419754862338746152976542491519424567Nov 18 2001
18519835647242376815984057156965175048Nov 19 2001
18619867235443169582785766026162066758Nov 19 2001
18719867542334615978268773441114437786Nov 19 2001
18821367485937645891280439804940893408Nov 19 2001
18921387546926379148556418527572581465Nov 19 2001
19021478569335827461976952262326225967Nov 22 2001
19121598746332617548970449816361894407Nov 24 2001
19221647583929673851464236718781763246Nov 25 2001
19321834576938615749284315854545851348Nov 26 2001
19421953674831649875269483106760138496Nov 27 2001
19521953867428937516463529039793092536Nov 27 2001
19621957438631479258669120388788302196Nov 27 2001
19723419867524539681757471609390617475Nov 30 2001
19823594178627951834665950057775005956Dec 4 2001
19923695718428576931467715091919051776Dec 5 2001
20023741589634695721882373158785137328Dec 6 2001
20123791685435981642785606392329360658Dec 7 2001
20223948576125391784660809708580790806Dec 15 2001
20323951684727196354865139851515893156Dec 15 2001
20424196873524681539759721609390612795Dec 16 2001
20524315768927516389466908216561280966Dec 16 2001
20624589736131258694776864305350346867Dec 20 2001
20724659831736287519489484412121448498Dec 22 2001
20824671953834152879684261826762816248Dec 22 2001
20924871365926391578465639460306493656Dec 23 2001
21024918576335429817688286061316068288Dec 23 2001
21125349687128691453772731937373913727Dec 26 2001
21226795348126831945771897132523179817Jan 7 2002
21326874915328143576975635624542653657Jan 8 2002
21427165384935942867197640181918104679Jan 9 2002
21527341685928675149378402692529620487Jan 10 2002
21627354689132549167889037236563273098Jan 12 2002
21727569184331852974687816052725061878Feb 6 2002
21828193465732415896791391647174619319Feb 7 2002
21929318564732516489795333680708633359Mar 3 2002
22029468573131472865992746044944064729Mar 4 2002
#Search ended on June 14, 2002.


Contributions

[ December 16, 2004 ]
Jean Claude Rosa (email) explains why there aren't any palindromic products of length 18.

A palindrome with 18 digits is always divisible by 11 and all
ninedigitals are divisible by 9 so a palindrome with 18 digits
as a product of two ninedigitals is divisible by 891 (9*9*11).
I verified that the number 10^18-1, composed of 18 digits '9'
is the sole 18-digit palindrome that is a multiple of 891
and this number is too large for being a product of two
ninedigitals.
Also if P is a palindrome with 18 digits (for example of
the form 1..... .....1) divisible by 891 then the difference
(10^18-1) - P is also a palindrome with 18 digits divisible
by 11 (thus of the form 8..... .....8) and so on...


Observations

One ninedigital only in the list can be multiplied
with two other ninedigitals to arrive at a palindrome.

172568349 * ↗
↘		213496587 = 36842753535724863 palindromic
      x
523876941 = 90404578787540409 palindromic

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

44589341514398544 = ↗
↘		139657428 * 319276548 N1 * N2 (64)
      x                x
159638274 * 279314856 N2/2 * N1*2 (119)
64236718781763246 = ↗
↘		148369257 * 432951678 N1 * N2 (86)
      x                x
216475839 * 296738514 N2/2 * N1*2 (192)
67973601310637976 = ↗
↘		173629458 * 391486572 N1 * N2 (141)
      x                x
195743286 * 347258916 N2/2 * N1*2 (179)

Two palindromes lead us through a waterfall of
transformations to the topic of pandigital numbers
in this case as products of two 5-digit factors
(Resp. index numbers 14 and 17 from the above list).

87734550305543778
=
697582431
*
125769438
⇊
[ 10-complement ]
⇊
985341672
=
13986 * 70452		   44127876367872144
=
349251768
*
126349758
⇊
[ 10-complement ]
⇊
984761352
=
21837 * 45096
 

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