HOME plateWON | World!OfNumbers Palindromic Productsof Two Ninedigital Numbers 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

 # Multiplicandsorted by Multiplier Palindromic product Date 1 123456789 538721694 66508850505880566 Apr 09 2001 2 123586749 571892643 70678352525387607 Sep 23 2001 3 123795864 192467538 23826685158662832 Sep 26 2001 4 124387569 137682459 17125986368952171 Sep 27 2001 5 124579368 645739281 80445791519754408 Sep 27 2001 6 124639578 236579814 29487208180278492 Sep 27 2001 7 124653879 328145976 40904668786640904 Sep 27 2001 8 124785693 273984651 34189364546398143 Sep 27 2001 9 124859376 189754632 23692644944629632 Sep 27 2001 10 124863795 429687153 53652368586325635 Sep 27 2001 11 124956738 518924367 64843096169034846 Sep 27 2001 12 125347689 391827546 49114677377641194 Sep 27 2001 13 125498736 357841629 44908672127680944 Sep 27 2001 14 125769438 697582431 87734550305543778 Sep 27 2001 15 125843697 587643921 73951283538215937 Sep 28 2001 16 125946738 495163827 62364268786246326 Sep 28 2001 17 126349758 349251768 44127876367872144 Sep 28 2001 18 126457983 561374829 70990328582309907 Sep 28 2001 19 126473589 249638517 31572679197627513 Sep 28 2001 20 126847539 568247193 72080757975708027 Sep 28 2001 21 127395468 218693754 27860593139506872 Sep 28 2001 22 127485369 156827439 19993203930239991 Sep 29 2001 23 127684953 548762319 70068690909686007 Sep 29 2001 24 127864593 612743859 78348244144284387 Sep 29 2001 25 127963584 347952816 44525289398252544 Sep 29 2001 26 128349576 378214659 48543691119634584 Sep 29 2001 27 128465937 427319685 54896023732069845 Sep 29 2001 28 128739564 328647591 42309947574990324 Sep 29 2001 29 128796453 497156382 64031978587913046 Sep 29 2001 30 129354786 361879524 46810848384801864 Sep 30 2001 31 129534687 589327641 76338371517383367 Sep 30 2001 32 129745386 463897521 60188562926588106 Sep 30 2001 33 129834657 234816579 30487329992378403 Sep 30 2001 34 132458967 589274361 78054673137645087 Sep 30 2001 35 132547986 346571289 45937326362373954 Sep 30 2001 36 132598647 672815934 89214482528441298 Sep 30 2001 37 132645897 514297638 68219471517491286 Sep 30 2001 38 132854679 512867934 68136904740963186 Sep 30 2001 39 132875649 729345681 96912280708221969 Oct 1 2001 40 132945867 175842396 23377519791577332 Oct 1 2001 41 134857269 394672815 53224497979442235 Oct 1 2001 42 134857692 342158967 46142768586724164 Oct 1 2001 43 134952687 197835426 26698422322489662 Oct 2 2001 44 134982576 361274859 48765811111856784 Oct 2 2001 45 135248967 582379461 78766220502266787 Oct 2 2001 46 135268947 381296745 51577609190677515 Oct 2 2001 47 135692478 357619428 48526266366262584 Oct 2 2001 48 135729648 469382157 63709074947090736 Oct 2 2001 49 135946872 619752834 84253459195435248 Oct 3 2001 50 135962487 271658349 36935344744353963 Oct 3 2001 51 136259487 471962358 64309348784390346 Oct 3 2001 52 136472985 426735981 58237933133973285 Oct 3 2001 53 137254896 493652781 67756261116265776 Oct 4 2001 54 137542968 168935274 23235858985853232 Oct 4 2001 55 137984652 476829153 65795104740159756 Oct 5 2001 56 138429675 418632597 57951174347115975 Oct 5 2001 57 138456927 238615479 33037965956973033 Oct 5 2001 58 138526947 495276138 68609091319090686 Oct 6 2001 59 138796254 156437928 21712998389921712 Oct 6 2001 60 138976245 378261945 52569424742496525 Oct 6 2001 61 139285647 491265738 68426266166262486 Oct 6 2001 62 139457286 438971256 61217739993771216 Oct 6 2001 63 139526487 187439526 26152778587725162 Oct 6 2001 64 139657428 319276548 44589341514398544 Oct 8 2001 65 139765248 593142786 82900748584700928 Oct 8 2001 66 139867245 358167249 50095866366859005 Oct 8 2001 67 142357689 179524368 25556674147665552 Oct 8 2001 68 142685397 548923671 78323391919332387 Oct 8 2001 69 142937856 178654392 25536475757463552 Oct 9 2001 70 143682579 294738156 42348738383784324 Oct 9 2001 71 145238796 465923781 67670208980207676 Oct 11 2001 72 145287369 196543278 28555255755255582 Oct 11 2001 73 145689237 218537649 31838583338583813 Oct 11 2001 74 145862739 194652738 28392581518529382 Oct 12 2001 75 145967328 279435618 40788470507488704 Oct 12 2001 76 146279583 526137849 76963225152236967 Oct 12 2001 77 146573289 251893467 36920853935802963 Oct 13 2001 78 146735892 421653798 61871746164717816 Oct 13 2001 79 146758392 457369128 67122757775722176 Oct 13 2001 80 147239568 574821936 84636533533563648 Oct 13 2001 81 147268395 372854691 54909711911790945 Oct 13 2001 82 147629538 154378269 22790792529709722 Oct 14 2001 83 147653289 378241695 55848630303684855 Oct 14 2001 84 147965238 576938241 85366804140866358 Oct 14 2001 85 148297356 315482769 46785260506258764 Oct 14 2001 86 148369257 432951678 64236718781763246 Oct 14 2001 87 148627593 615279483 91447508580574419 Oct 14 2001 88 148769235 389167425 57896140104169875 Oct 15 2001 89 148965237 241375869 35956613531665953 Oct 15 2001 90 149273856 168539742 25158577177585152 Oct 15 2001 91 149536287 641329857 95902085558020959 Oct 16 2001 92 149573268 437586192 65451196769115456 Oct 16 2001 93 149652378 273415968 40917349794371904 Oct 16 2001 94 149672583 578431296 86575306160357568 Oct 16 2001 95 152394678 531429786 80987071117078908 Oct 18 2001 96 152396784 568931427 86703319791330768 Oct 18 2001 97 152473986 431852796 65846317171364856 Oct 18 2001 98 152697348 563427891 86033944744933068 Oct 19 2001 99 152768934 326748591 49917033933071994 Oct 19 2001 100 152784936 451782936 69025626962652096 Oct 19 2001 101 152796483 256498371 39192048984029193 Oct 19 2001 102 152847936 179634852 27456816361865472 Oct 20 2001 103 152946378 432156897 66096832123869066 Oct 20 2001 104 152983467 258346179 39522694149622593 Oct 20 2001 105 153869427 465832971 71677452325477617 Oct 20 2001 106 154392786 165478932 25548753335784552 Oct 21 2001 107 154837629 496718523 76910718381701967 Oct 21 2001 108 154872963 381749265 59122639793622195 Oct 21 2001 109 156738924 513829467 80537077777073508 Oct 22 2001 110 157328946 427813956 67307518781570376 Oct 23 2001 111 157329648 186729354 29378063536087392 Oct 23 2001 112 157426839 367489125 57852651315625875 Oct 24 2001 113 157639824 297413586 46884225352248864 Oct 24 2001 114 158263749 458691723 72594271717249527 Oct 25 2001 115 158269743 481257369 76168480108486167 Oct 25 2001 116 158274693 569824713 90188831513888109 Oct 25 2001 117 159246783 386174952 61497118781179416 Oct 27 2001 118 159428736 387942156 61849127572194816 Oct 27 2001 119 159638274 279314856 44589341514398544 Oct 27 2001 120 162438759 275984316 44830549794503844 Oct 27 2001 121 162735489 328467915 53453386768335435 Oct 27 2001 122 162937845 348926517 56853334743335865 Oct 28 2001 123 163759824 295473816 48386740104768384 Oct 28 2001 124 163795842 376592148 61684227972248616 Oct 28 2001 125 163874529 485217693 79514820902841597 Oct 28 2001 126 164875293 387614952 63908128782180936 Oct 28 2001 127 165273894 289345671 47821285758212874 Oct 29 2001 128 167258349 367125984 61404885958840416 Oct 29 2001 129 167259384 265387941 44388623532688344 Oct 29 2001 130 167529834 374251689 62698323332389626 Oct 29 2001 131 167953248 491582736 82562917171926528 Oct 29 2001 132 168473259 493765182 83186229392268138 Oct 29 2001 133 168597324 387421569 65318239793281356 Oct 29 2001 134 169247538 391486257 66258085158085266 Oct 30 2001 135 172369854 367512984 63348159395184336 Oct 30 2001 136 172568349 213496587 36842753535724863 Oct 30 2001 137 172568349 523876941 90404578787540409 Oct 30 2001 138 172698354 279318546 48237853135873284 Oct 30 2001 139 172983546 273684519 47342918581924374 Oct 30 2001 140 173542986 375291846 65129267576292156 Oct 30 2001 141 173629458 391486572 67973601310637976 Oct 30 2001 142 173894265 321685947 55939341314393955 Oct 31 2001 143 173952468 234165978 40733749794733704 Oct 31 2001 144 174283695 318627549 55531586568513555 Oct 31 2001 145 174985623 561798243 98306615551660389 Oct 31 2001 146 174986325 327648195 57333953535933375 Oct 31 2001 147 175263894 367182459 64353827572835346 Oct 31 2001 148 175298643 451623879 79169053135096197 Oct 31 2001 149 175386249 358219674 62826804940862826 Oct 31 2001 150 175648329 317682495 55800399399300855 Oct 31 2001 151 175694283 381456972 67019809190891076 Oct 31 2001 152 175849236 394287561 69335166366153396 Oct 31 2001 153 175984326 475918263 83754154745145738 Oct 31 2001 154 178592463 485912736 86780352325308768 Nov 4 2001 155 179238456 462798513 82951290909215928 Nov 4 2001 156 179843265 321469875 57814191919141875 Nov 4 2001 157 179852634 257348196 46284750905748264 Nov 4 2001 158 182469375 285746139 52139919391993125 Nov 5 2001 159 182793465 287163945 52491692529619425 Nov 6 2001 160 182967543 314968725 57629053735092675 Nov 6 2001 161 183542697 518639247 95192446164429159 Nov 6 2001 162 183695274 348625179 64040797779704046 Nov 7 2001 163 184523679 375618924 69310585758501396 Nov 8 2001 164 184537629 436897152 80623964546932608 Nov 8 2001 165 184752693 417623589 77157082728075177 Nov 9 2001 166 185249736 375942861 69643315751334696 Nov 9 2001 167 186279453 345169782 64298038183089246 Nov 10 2001 168 187539462 465728319 87342438383424378 Nov 11 2001 169 189367542 428153679 81078409790487018 Nov 11 2001 170 192457386 439275618 84541837173814548 Nov 12 2001 171 192476853 281374965 54158167776185145 Nov 12 2001 172 192746385 279845631 53939233733293935 Nov 13 2001 173 193476582 256438917 49614925152941694 Nov 14 2001 174 193872465 281697453 54613379597331645 Nov 15 2001 175 194287563 472698513 91839442124493819 Nov 15 2001 176 194675328 316894572 61691554745519616 Nov 16 2001 177 195364287 493621587 96436029392063469 Nov 16 2001 178 195472368 413827956 80891930503919808 Nov 17 2001 179 195743286 347258916 67973601310637976 Nov 17 2001 180 196254837 289731645 56861236763216865 Nov 17 2001 181 196728345 281367459 55352954545925355 Nov 18 2001 182 196784352 214569387 42223897779832224 Nov 18 2001 183 196823457 346721598 68242943534924286 Nov 18 2001 184 197548623 387461529 76542491519424567 Nov 18 2001 185 198356472 423768159 84057156965175048 Nov 19 2001 186 198672354 431695827 85766026162066758 Nov 19 2001 187 198675423 346159782 68773441114437786 Nov 19 2001 188 213674859 376458912 80439804940893408 Nov 19 2001 189 213875469 263791485 56418527572581465 Nov 19 2001 190 214785693 358274619 76952262326225967 Nov 22 2001 191 215987463 326175489 70449816361894407 Nov 24 2001 192 216475839 296738514 64236718781763246 Nov 25 2001 193 218345769 386157492 84315854545851348 Nov 26 2001 194 219536748 316498752 69483106760138496 Nov 27 2001 195 219538674 289375164 63529039793092536 Nov 27 2001 196 219574386 314792586 69120388788302196 Nov 27 2001 197 234198675 245396817 57471609390617475 Nov 30 2001 198 235941786 279518346 65950057775005956 Dec 4 2001 199 236957184 285769314 67715091919051776 Dec 5 2001 200 237415896 346957218 82373158785137328 Dec 6 2001 201 237916854 359816427 85606392329360658 Dec 7 2001 202 239485761 253917846 60809708580790806 Dec 15 2001 203 239516847 271963548 65139851515893156 Dec 15 2001 204 241968735 246815397 59721609390612795 Dec 16 2001 205 243157689 275163894 66908216561280966 Dec 16 2001 206 245897361 312586947 76864305350346867 Dec 20 2001 207 246598317 362875194 89484412121448498 Dec 22 2001 208 246719538 341528796 84261826762816248 Dec 22 2001 209 248713659 263915784 65639460306493656 Dec 23 2001 210 249185763 354298176 88286061316068288 Dec 23 2001 211 253496871 286914537 72731937373913727 Dec 26 2001 212 267953481 268319457 71897132523179817 Jan 7 2002 213 268749153 281435769 75635624542653657 Jan 8 2002 214 271653849 359428671 97640181918104679 Jan 9 2002 215 273416859 286751493 78402692529620487 Jan 10 2002 216 273546891 325491678 89037236563273098 Jan 12 2002 217 275691843 318529746 87816052725061878 Feb 6 2002 218 281934657 324158967 91391647174619319 Feb 7 2002 219 293185647 325164897 95333680708633359 Mar 3 2002 220 294685731 314728659 92746044944064729 Mar 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      x523876941 = 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                x159638274 * 279314856 N2/2 * N1*2 (119)

 64236718781763246 = ↗↘ 148369257 * 432951678 N1 * N2 (86)      x                x216475839 * 296738514 N2/2 * N1*2 (192)

 67973601310637976 = ↗↘ 173629458 * 391486572 N1 * N2 (141)      x                x195743286 * 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