## Cubes with same digits

The digits of
$5^3 \; = \; 125$
can be permuted to form
$8^3 \; = \; 512$

cube whose digits can be permuted to produce two other cubes :

$345^3 \; = \; 41063625$
$384^3 \; = \; 56623104$
$405^3 \; = \; 66430125$

$1001^3 \; = \; 1003003001$
$1010^3 \; = \; 1030301000$
$1100^3 \; = \; 1331000000$

$1028^3 \; = \; 1086373952$
$1112^3 \; = \; 1375036928$
$1721^3 \; = \; 5097328361$

$1088^3 \; = \; 1287913472$
$1238^3 \; = \; 1897413272$
$1571^3 \; = \; 3877292411$

$1273^3 \; = \; 2062933417$
$1447^3 \; = \; 3029741623$
$2101^3 \; = \; 9274236301$

$1532^3 \; = \; 3595640768$
$1889^3 \; = \; 6740558369$
$1919^3 \; = \; 7066834559$

$1733^3 \; = \; 5204699837$
$1895^3 \; = \; 6804992375$
$2033^3 \; = \; 8402569937$

and three other cubes :

$1002^3 \; = \; 1006012008$
$1020^3 \; = \; 1061208000$
$2001^3 \; = \; 8012006001$
$2010^3 \; = \; 8120601000$

Paul found:

$10001^3 \; = \; 1000300030001$
$10010^3 \; = \; 1003003001000$
$10100^3 \; = \; 1030301000000$
$11000^3 \; = \; 1331000000000$

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

$10003^3 \; = \; 1000900270027$
$10030^3 \; = \; 1009027027000$
$10300^3 \; = \; 1092727000000$

$10004^3 \; = \; 1001200480064$
$10040^3 \; = \; 1012048064000$
$10400^3 \; = \; 1124864000000$

$10017^3 \; = \; 1005108674913$
$14589^3 \; = \; 3105107018469$
$17001^3 \; = \; 4913867051001$

$10018^3 \; = \; 1005409725832$
$17650^3 \; = \; 5498372125000$
$18001^3 \; = \; 5832972054001$
$20419^3 \; = \; 8513407220059$

$10047^3 \; = \; 1014166373823$
$13896^3 \; = \; 2683301147136$
$19146^3 \; = \; 7018336124136$

$10059^3 \; = \; 1017804635379$
$12117^3 \; = \; 1779038405613$
$17883^3 \; = \; 5719013604387$

$10093^3 \; = \; 1028160274357$
$10267^3 \; = \; 1082257704163$
$13027^3 \; = \; 2210717450683$

math grad - Interest: Number theory
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### 2 Responses to Cubes with same digits

1. Paul says:

Here are a few 5 digit ones, notice how some n and n^3 each contain the same digits.
I have a feeling there will be a lot of 5 digit variants.

{10001,^3 -> ,10001,^3, = ,1000300030001}
{10001,^3 -> ,10010,^3, = ,1003003001000}
{10001,^3 -> ,10100,^3, = ,1030301000000}
{10001,^3 -> ,11000,^3, = ,1331000000000}

{10002,^3 -> ,10002,^3, = ,1000600120008}
{10002,^3 -> ,10020,^3, = ,1006012008000}
{10002,^3 -> ,10200,^3, = ,1061208000000}
{10002,^3 -> ,20001,^3, = ,8001200060001}
{10002,^3 -> ,20010,^3, = ,8012006001000}
{10002,^3 -> ,20100,^3, = ,8120601000000}

{10003,^3 -> ,10003,^3, = ,1000900270027}
{10003,^3 -> ,10030,^3, = ,1009027027000}
{10003,^3 -> ,10300,^3, = ,1092727000000}

{10004,^3 -> ,10004,^3, = ,1001200480064}
{10004,^3 -> ,10040,^3, = ,1012048064000}
{10004,^3 -> ,10400,^3, = ,1124864000000}

{10010,^3 -> ,10001,^3, = ,1000300030001}
{10010,^3 -> ,10010,^3, = ,1003003001000}
{10010,^3 -> ,10100,^3, = ,1030301000000}
{10010,^3 -> ,11000,^3, = ,1331000000000}

{10017,^3 -> ,10017,^3, = ,1005108674913}
{10017,^3 -> ,14589,^3, = ,3105107018469}
{10017,^3 -> ,17001,^3, = ,4913867051001}

{10018,^3 -> ,10018,^3, = ,1005409725832}
{10018,^3 -> ,17650,^3, = ,5498372125000}
{10018,^3 -> ,18001,^3, = ,5832972054001}
{10018,^3 -> ,20419,^3, = ,8513407220059}

{10020,^3 -> ,10002,^3, = ,1000600120008}
{10020,^3 -> ,10020,^3, = ,1006012008000}
{10020,^3 -> ,10200,^3, = ,1061208000000}
{10020,^3 -> ,20001,^3, = ,8001200060001}
{10020,^3 -> ,20010,^3, = ,8012006001000}
{10020,^3 -> ,20100,^3, = ,8120601000000}

{10030,^3 -> ,10003,^3, = ,1000900270027}
{10030,^3 -> ,10030,^3, = ,1009027027000}
{10030,^3 -> ,10300,^3, = ,1092727000000}

{10040,^3 -> ,10004,^3, = ,1001200480064}
{10040,^3 -> ,10040,^3, = ,1012048064000}
{10040,^3 -> ,10400,^3, = ,1124864000000}

{10047,^3 -> ,10047,^3, = ,1014166373823}
{10047,^3 -> ,13896,^3, = ,2683301147136}
{10047,^3 -> ,19146,^3, = ,7018336124136}

{10059,^3 -> ,10059,^3, = ,1017804635379}
{10059,^3 -> ,12117,^3, = ,1779038405613}
{10059,^3 -> ,17883,^3, = ,5719013604387}

{10093,^3 -> ,10093,^3, = ,1028160274357}
{10093,^3 -> ,10267,^3, = ,1082257704163}
{10093,^3 -> ,13027,^3, = ,2210717450683}

{10100,^3 -> ,10001,^3, = ,1000300030001}
{10100,^3 -> ,10010,^3, = ,1003003001000}
{10100,^3 -> ,10100,^3, = ,1030301000000}
{10100,^3 -> ,11000,^3, = ,1331000000000}

Paul.