Some solutions to Project Euler 62: Cubic permutations

 
Project Euler 62
https://projecteuler.net/index.php?section=problems&id=62

The cube,   41063625 \; = \; 345^3,   can be permuted to produce two other cubes:

56623104 \; = \; 384^3    and
66430125 \; = \; 405^3

In fact,   41063625   is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

 

                ****************************************       

 

Here are some solutions:
 

The 12-digit cube   127035954683   has four other permutations that are cubes:

127035954683 \; = \; 5027^3
352045367981 \; = \; 7061^3
373559126408 \; = \; 7202^3
569310543872 \; = \; 8288^3
589323567104 \; = \; 8384^3

 
The next set is :

1402837695364 \; = \; 5196^3
5361789306244 \; = \; 8124^3
6132584079364 \; = \; 8496^3
9132376564084 \; = \; 9702^3
9363024516874 \; = \; 9783^3

 

The 13-digit cube   1961574655832   has four other permutations that are cubes:

1961574655832 \; = \; 12518^3
2981631556457 \; = \; 14393^3
5657831164259 \; = \; 17819^3
8631525145697 \; = \; 20513^3
9625537115648 \; = \; 21272^3

 
The next set is :

2097643558401 \; = \; 12801^3
2501609484375 \; = \; 13575^3
2595641083407 \; = \; 13743^3
6140550947328 \; = \; 18312^3
9083540714625 \; = \; 20865^3

 
Then,

2546097986375 \; = \; 13655^3
2567976059384 \; = \; 13694^3
5429503678976 \; = \; 17576^3 \; = \; 26^9
6587579024936 \; = \; 18746^3
7736546098952 \; = \; 19778^3

 

The 13-digit cube   1000600120008   has five other permutations that are cubes:

1000600120008 \; = \; 10002^3
1006012008000 \; = \; 10020^3
1061208000000 \; = \; 10200^3
8001200060001 \; = \; 20001^3
8012006001000 \; = \; 20010^3
8120601000000 \; = \; 20100^3

 
The next set is :

1426487591593 \; = \; 11257^3
1432197595648 \; = \; 11272^3
3496581419752 \; = \; 15178^3
4275981654391 \; = \; 16231^3
4813967954125 \; = \; 16885^3
7591941538264 \; = \; 19654^3

 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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4 Responses to Some solutions to Project Euler 62: Cubic permutations

  1. paul says:

    !4 digit cubes starting with 6 permutations, then 7, 8 and 9.

    6 Perms

    11064224539187 = 22283^3
    14679130284125 = 24485^3
    16480142715392 = 25448^3
    19322185461704 = 26834^3
    32464811925701 = 31901^3
    46512291801347 = 35963^3

    11228890332457 = 22393^3
    27590148228313 = 30217^3
    28302913421875 = 30475^3
    32425138819072 = 31888^3
    32820251748913 = 32017^3
    87534298213201 = 44401^3

    11824302795768 = 22782^3
    14637802579128 = 24462^3
    21563878129704 = 27834^3
    21780751246983 = 27927^3
    22771068931584 = 28344^3
    90488612217375 = 44895^3

    12297598507368 = 23082^3
    29857358209167 = 31023^3
    55227991087368 = 38082^3
    69517788295032 = 41118^3
    75735288109629 = 42309^3
    82876153250799 = 43599^3

    12435568197632 = 23168^3
    23654732519816 = 28706^3
    28557213361496 = 30566^3
    32685125964317 = 31973^3
    48639116235527 = 36503^3
    63832146957125 = 39965^3

    12852368769024 = 23424^3
    21738662850429 = 27909^3
    23472268051968 = 28632^3
    27048629165832 = 30018^3
    43822698057216 = 35256^3
    52242319660887 = 37383^3

    14290527842163 = 24267^3
    14827105492263 = 24567^3
    15482932761024 = 24924^3
    22742159410368 = 28332^3
    42286042531719 = 34839^3
    48022145376192 = 36348^3

    14861533742056 = 24586^3
    15845403276613 = 25117^3
    41013654783256 = 34486^3
    45331267580416 = 35656^3
    60814367453125 = 39325^3
    75162133546408 = 42202^3

    15282532970496 = 24816^3
    20453927568192 = 27348^3
    36192428790552 = 33078^3
    38157029452296 = 33666^3
    38906527429125 = 33885^3
    86345027212599 = 44199^3

    17859401723619 = 26139^3
    21591780491736 = 27846^3
    24979031175168 = 29232^3
    34917687019125 = 32685^3
    60879341715219 = 39339^3
    97196410781352 = 45978^3

    20429256361753 = 27337^3
    32516740952632 = 31918^3
    35365069172224 = 32824^3
    37209526341625 = 33385^3
    53259231706624 = 37624^3
    92063755312264 = 45154^3

    20645238791737 = 27433^3
    22831375767409 = 28369^3
    30234279618577 = 31153^3
    57146703927832 = 38518^3
    83327476901752 = 43678^3
    90367732421875 = 44875^3

    24469825120768 = 29032^3
    42082459126687 = 34783^3
    42640229817856 = 34936^3
    42816226875904 = 34984^3
    52827241960648 = 37522^3
    88174602564229 = 44509^3

    25979045828113 = 29617^3
    31590457812928 = 31612^3
    32415988079125 = 31885^3
    52839914078125 = 37525^3
    58451728309129 = 38809^3
    87109155423289 = 44329^3

    26615737485793 = 29857^3
    38672765749513 = 33817^3
    49176737526853 = 36637^3
    65816927734375 = 40375^3
    68297557341376 = 40876^3
    79257646133875 = 42955^3

    38187639421875 = 33675^3
    38197846381752 = 33678^3
    38813597864712 = 33858^3
    69837818532741 = 41181^3
    71623315478889 = 41529^3
    86239578381741 = 44181^3

    7 perms

    10569784298536 = 21946^3
    18083756564992 = 26248^3
    18965807549632 = 26668^3
    50628361995784 = 36994^3
    75966439518208 = 42352^3
    97831984066552 = 46078^3
    99852138660457 = 46393^3

    10849635532072 = 22138^3
    21559230074368 = 27832^3
    24538160052379 = 29059^3
    25523849601307 = 29443^3
    40725336980125 = 34405^3
    41260329503875 = 34555^3
    81503765420293 = 43357^3

    12356831794159 = 23119^3
    12938153594176 = 23476^3
    31491628517539 = 31579^3
    35219817465139 = 32779^3
    47198521353619 = 36139^3
    81469933172551 = 43351^3
    96557281941133 = 45877^3

    12804692354875 = 23395^3
    13097526845248 = 23572^3
    15490388426752 = 24928^3
    16443257028589 = 25429^3
    17205482538496 = 25816^3
    38405782562419 = 33739^3
    42607284155983 = 34927^3

    15354698826375 = 24855^3
    15696358354872 = 25038^3
    35197258563648 = 32772^3
    55659832683741 = 38181^3
    56689485123375 = 38415^3
    58596436857321 = 38841^3
    97825614658533 = 46077^3

    16577473805992 = 25498^3
    34776859950721 = 32641^3
    50714629978375 = 37015^3
    70578124593976 = 41326^3
    75998730546712 = 42358^3
    79856973251047 = 43063^3
    79923751046875 = 43075^3

    16708497237125 = 25565^3
    25981677417032 = 29618^3
    29432076171875 = 30875^3
    47638707921251 = 36251^3
    78501712967432 = 42818^3
    86427102371597 = 44213^3
    91283041277576 = 45026^3

    8 perms

    10314675896832 = 21768^3
    12417863650893 = 23157^3
    13974881626503 = 24087^3
    19317865402368 = 26832^3
    71096865321483 = 41427^3
    78342316815069 = 42789^3
    89260685431371 = 44691^3
    98534308116672 = 46188^3

    10340284735656 = 21786^3
    18437405506632 = 26418^3
    26055434078136 = 29646^3
    35245610867403 = 32787^3
    36015465037824 = 33024^3
    36310726085544 = 33114^3
    43410085673625 = 35145^3
    65724053438016 = 40356^3

    10418796526321 = 21841^3
    10591472326681 = 21961^3
    20194756231168 = 27232^3
    27328019461561 = 30121^3
    43191827216056 = 35086^3
    61051282149376 = 39376^3
    74826025131619 = 42139^3
    90567241861312 = 44908^3

    12620043581768 = 23282^3
    16602842507831 = 25511^3
    16702615804328 = 25562^3
    18618053272064 = 26504^3
    23102857608641 = 28481^3
    25207880643161 = 29321^3
    63287460812051 = 39851^3
    68724510023681 = 40961^3

    15302864497283 = 24827^3
    22380469853741 = 28181^3
    25084283369417 = 29273^3
    42064313878952 = 34778^3
    42930148628375 = 35015^3
    72490848321536 = 41696^3
    82562904733184 = 43544^3
    85340912247683 = 44027^3

    9 Perms

    13465983902671 = 23791^3
    18102364796593 = 26257^3
    23667095189431 = 28711^3
    23719065439168 = 28732^3
    38160429751963 = 33667^3
    68312596071439 = 40879^3
    68523370149961 = 40921^3
    80997264315163 = 43267^3
    96803741135296 = 45916^3

    Paul.

  2. paul says:

    15 Digits cubes have quite a lot, so here are those with 12 permutations

    106739258417856 = 47436^3
    109785265614387 = 47883^3
    114876307556928 = 48612^3
    131856684750792 = 50898^3
    169048165238757 = 55293^3
    183066597827541 = 56781^3
    186453210757689 = 57129^3
    214190675578368 = 59832^3
    669531482107587 = 87483^3
    674018635189752 = 87678^3
    765458231098176 = 91476^3
    885731701294656 = 96036^3

    112706583998464 = 48304^3
    151436482979608 = 53302^3
    164368470918952 = 54778^3
    198124698564073 = 58297^3
    204117986685439 = 58879^3
    364928067945181 = 71461^3
    396401268841579 = 73459^3
    416548203169987 = 74683^3
    572696830149184 = 83044^3
    841686945209713 = 94417^3
    864597134281096 = 95266^3
    950267986183144 = 98314^3

    and 11

    101729840326568 = 46682^3
    104680231527689 = 47129^3
    136781926005824 = 51524^3
    210528068347169 = 59489^3
    304712526961088 = 67292^3
    319200124657688 = 68342^3
    502668472109831 = 79511^3
    598672128360041 = 84281^3
    625860113940872 = 85538^3
    828147650069312 = 93908^3
    840751262310968 = 94382^3

    104573652871897 = 47113^3
    128948357570761 = 50521^3
    147055817932687 = 52783^3
    158472163970875 = 54115^3
    204867717859351 = 58951^3
    250487817791653 = 63037^3
    341810677278559 = 69919^3
    397470681578125 = 73525^3
    679731084578125 = 87925^3
    739867054871125 = 90445^3
    788914637557201 = 92401^3

    125631058992704 = 50084^3
    212060594589137 = 59633^3
    239540911807625 = 62105^3
    316851909572024 = 68174^3
    405865029721319 = 74039^3
    516100498223597 = 80213^3
    524250018996137 = 80633^3
    692003421597851 = 88451^3
    904932561078125 = 96725^3
    930428791015625 = 97625^3
    973802164502159 = 99119^3

    126120833457949 = 50149^3
    127369820359144 = 50314^3
    129523494108736 = 50596^3
    196538434097221 = 58141^3
    223947119460583 = 60727^3
    269811549437032 = 64618^3
    297061398351424 = 66724^3
    304698942523171 = 67291^3
    340229194658731 = 69811^3
    341019324789625 = 69865^3
    803449196732125 = 92965^3

    129308576850432 = 50568^3
    186042283957503 = 57087^3
    239552483087016 = 62106^3
    259633020184875 = 63795^3
    267513238098504 = 64434^3
    401898256032573 = 73797^3
    500698587232143 = 79407^3
    512826043839507 = 80043^3
    545679003228813 = 81717^3
    800859237314625 = 92865^3
    901260738352584 = 96594^3

    Paul.

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