## a^3 + b^3 + c^3 = d^3 + e^3 + f^3 and a*b*c = d*e*f

If    $a^3 \; + \; b^3 \; = \; c^3 \; + \; d^3$
then    $(a \, c)^3 \; + \; (b \, c)^3 \; + \; (d^2)^3 \; = \; (a \, d)^3 \; + \; (b \, d)^3 \; + \; (c^2)^3$
and    $(a \, c) \,(b \, c)(d^2) \; = \; (a \, b) \, c^2 \, d^2 \; = \; (a \, d) \,(b \, d) \,(c^2)$

$a^3 \; + \; b^3 \; - \; c^3 \; = \; d^3$
$a^3 \; + \; b^3 \; - \; d^3 \; = \; c^3$

$(a \, c)^3 \; + \; (b \, c)^3 \; + \; (d^2)^3$
$= \; a^3 \, c^3 \; + \; b^3 \, c^3 \; + \; d^6$
$= \; c^3 \,(a^3 + b^3) \; + \; d^6$
$= \; (a^3 + b^3 - d^3) \,(a^3 + b^3) \; + \; d^6$
$= \; a^6 \; + \; 2 \, a^3 \, b^3 \; - \; a^3 \, d^3 \; + \; b^6 \; - \; b^3 \, d^3 \; + \; d^6$   ….. (1)

$(a \, d)^3 \; + \; (b \, d)^3 \; + \; (c^2)^3$
$= \; a^3 \, d^3 \; + \; b^3 \, d^3 \; + \; c^6$
$= \; d^3 \,(a^3 + b^3) \; + \; c^6$
$= \; (a^3 + b^3 - c^3) \,(a^3 + b^3) \; + \; c^6$
$= \; a^6 \; + \; 2 \, a^3 \, b^3 \; - \; a^3 \, c^3 \; + \; b^6 \; - \; b^3 \, c^3 \; + \; c^6$   ….. (2)

(1)   –   (2) :
$= \; a^3 \, c^3 \; - \; a^3 \, d^3 \; + \; b^3 \, c^3 \; - \; b^3 \, d^3 \; - \; c^6 \; + \; d^6$
$= \; (a^3 + b^3) \, (c^3 - d^3) \; - \; c^6 \; + \; d^6$
since    $a^3 \; + \; b^3 \; = \; c^3 \; + \; d^3$
$= \; (c^3 + d^3) \,(c^3 - d^3) \; - \; c^6 \; + \; d^6$
$= \; 0$

then    $(a \, c)^3 \; + \; (b \, c)^3 \; + \; (d^2)^3 \; = \; (a \, d)^3 \; + \; (b \, d)^3 \; + \; (c^2)^3$

Solutions to   $a^3 + b^3 = c^3 + d^3$   are called “Ramanujan Numbers”
The first five of these are:
1729,    4104,    13832   (= 8 × 1729),    20683,    32832   (= 8 × 4104)

The sequence continues with
39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, …
OEIS A001235   https://oeis.org/A001235

Now, I’m going to use these   $N = a^3 + b^3 = c^3 + d^3$   “Ramanujan Numbers”
to write numbers expressible as the sum of three positive cubes in two different ways
such that

$M \; = \; A^3 + B^3 + C^3 \; = \; D^3 + E^3 + F^3$,    and
$A \cdot B \cdot C \; = \; D \cdot E \cdot F$

I ignore   13832   and   32832   since they are multiples

$1729 \; = 1^3 \; + \; 12^3 \; = \; 9^3 \; + \; 10^3$
$9^3 + 108^3 + 100^3 \; = \; 10^3 \; + \; 120^3 \; + \; 81^3 \; = \; 2260441$
$9\times 108\times 100 \; = \; 10\times 120\times 81 \; = \; 97200 \; = \; 12\times 90^2$

$4104 \; = \; 2^3 \; + \; 16^3 \; = \; 9^3 \; + \; 15^3$
$18^3 \; + \; 144^3 \; + \; 225^3 \; = \; 30^3 \; + \; 240^3 \; + \; 81^3 \; = \; 14382441$
$18\times 144\times 225 \; = \; 30\times 240\times 81 \; = \; 583200 \; = \; 32\times 135^2$

$20683 \; = \; 10^3 \; + \; 27^3 \; = \; 19^3 \; + \; 24^3$
$190^3 \; + \; 513^3 \; + \; 576^3 \; = \; 240^3 \; + \; 648^3 \; + \; 361^3 \; = \; 332967673$
$190\times 513\times 576 \; = \; 240\times 648\times 361 \; = \; 56142720 \; = \; 270\times 456^2$

Then,   I take the first triple pair solutions to   $a^3 + b^3 = c^3 + d^3 = e^3 + f^3$
which is the number,   87539319

$87539319 \; = \; 167^3 \; + \; 436^3 \; = \; 228^3 \; + \; 423^3 \; = \; 255^3 \; + \; 414^3$

(1)   $87539319 \; = \; 167^3 \; + \; 436^3 \; = \; 228^3 \; + \; 423^3$
(2)   $87539319 \; = \; 167^3 \; + \; 436^3 \; = \; 255^3 \; + \; 414^3$
(3)   $87539319 \; = \; 228^3 \; + \; 423^3 \; = \; 255^3 \; + \; 414^3$

(1)
$87539319 = 167^3 + 436^3 = 228^3 + 423^3$
$38076^3 + 99408^3 + 178929^3 = 70641^3 + 184428^3 + 51984^3 = 6766063796287377$
$38076\times 99408\times 178929 = 70641\times 184428\times 51984 = 677256823242432$

(2)
$87539319 = 167^3 + 436^3 = 255^3 + 414^3$
$42585^3 + 111180^3 + 171396^3 = 69138^3 + 180504^3 + 65025^3 = 6486552092290761$
$42585\times 111180\times 171396 = 69138\times 180504\times 65025 = 811491553018800$

(3)
$87539319 = 228^3 + 423^3 = 255^3 + 414^3$
$58140^3 + 107865^3 + 171396^3 = 94392^3 + 175122^3 + 65025^3 = 6486552092290761$
$58140\times 107865\times 171396 = 94392\times 175122\times 65025 = 1074870781455600$

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Here are the next numbers which are sum of 2 cubes in 3 different ways:

$119824488 = 11^3 + 493^3 = 90^3 + 492^3 = 346^3 + 428^3$
$143604279 = 111^3 + 522^3 = 408^3 + 423^3 = 359^3 + 460^3$
$175959000 = 70^3 + 560^3 = 198^3 + 552^3 = 315^3 + 525^3$
$327763000 = 339^3 + 661^3 = 300^3 + 670^3 = 510^3 + 580^3$

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The lowest number expressible as the sum of two positive cubes in 4 different ways is:

$6963472309248=13322^3+16630^3=10200^3+18072^3=5436^3+18948^3=2421^3+19083^3$

(1)   $6963472309248 = 13322^3 + 16630^3 = 10200^3 + 18072^3$
(2)   $6963472309248 = 13322^3 + 16630^3 = 5436^3 + 18948^3$
(3)   $6963472309248 = 13322^3 + 16630^3 = 2421^3 + 19083^3$
(4)   $6963472309248 = 10200^3 + 18072^3 = 5436^3 + 18948^3$
(5)   $6963472309248 = 10200^3 + 18072^3 = 2421^3 + 19083^3$
(6)   $6963472309248 = 5436^3 + 18948^3 = 2421^3 + 19083^3$

(1)
$6963472309248 = 13322^3 + 16630^3 = 10200^3 + 18072^3$

$135884400^3 + 169626000^3 + 326597184^3 = 240755184^3 + 300537360^3 + 104040000^3$
$= \; 42226416498575222162325504$

$135884400\times 169626000\times 326597184 = 240755184\times 300537360\times 104040000$
$= \; 7527910687286347929600000$

(2)
$6963472309248 = 13322^3 + 16630^3 = 5436^3 + 18948^3$

$72418392^3 + 90400680^3 + 359026704^3 = 252425256^3 + 315105240^3 + 29550096^3$
$= \; 47397177334862723948285952$

$72418392\times 90400680\times 359026704 = 252425256\times 315105240\times 29550096$
$= \; 2350430027714973450378240$

(3)
$6963472309248 = 13322^3 + 16630^3 = 2421^3 + 19083^3$

$32252562^3 + 40261230^3 + 364160889^3 = 254223726^3 + 317350290^3 + 5861241^3$
$= \; 48391335838652463435110697$

$32252562\times 40261230\times 364160889 = 254223726\times 317350290\times 5861241$
$= \; 472873044146651151250140$

(4)
$6963472309248 = 10200^3 + 18072^3 = 5436^3 + 18948^3$

$55447200^3 + 98239392^3 + 359026704^3 = 193269600^3 + 342428256^3 + 29550096^3$
$= \; 47397177334862723948285952$

$55447200\times 98239392\times 359026704 = 193269600\times 342428256\times 29550096$
$= \; 1955654077918228398489600$

(5)
$6963472309248 = 10200^3 + 18072^3 = 2421^3 + 19083^3$

$24694200^3 + 43752312^3 + 364160889^3 = 194646600^3 + 344867976^3 + 5861241^3$
$= \; 48391335838652463435110697$

$24694200\times 43752312\times 364160889 = 194646600\times 344867976\times 5861241$
$= \; 393449745884180982465600$

(6)
$6963472309248 = 5436^3 + 18948^3 = 2421^3 + 19083^3$

$13160556^3 + 45873108^3 + 364160889^3 = 103735188^3 + 361584684^3 + 5861241^3$
$= \; 48391335838652463435110697$

$13160556\times 45873108\times 364160889 = 103735188\times 361584684\times 5861241$
$= \; 219849612049260340914672$

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Here are the next numbers which are sum of 2 cubes in 4 different ways:

{12625136269928,   (12939,21869),   (10362,22580),   (7068,23066),   (4275,23237)}
{21131226514944,   (17176,25232),   (11772,26916),   (8664,27360),   (1539,27645)}
{26059452841000,   (21930,24940),   (14577,28423),   (12900,28810),   (4170,29620)}
{55707778473984,   (26644,33260),   (20400,36144),   (10872,37896),   (4842,38166)}

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Here are the first few numbers that can be formed by the sum of 2 cubes in 5 different ways:

48988659276962496:
{(231518,331954),(221424,336588),(205292,342952),(107839,362753),(38787,365757)}

391909274215699968 :
{(463036,663908),(442848,673176),(410584,685904),(215678,725506),(77574,731514)}

490593422681271000 :
{(579240,666630),(543145,691295),(285120,776070),(233775,781785),(48369,788631)}

1322693800477987392 :
{(694554,995862),(664272,1009764),(615876,1028856),(323517,1088259),(116361,1097271)}

3135274193725599744 :
{(926072,1327816),(885696,1346352),(821168,1371808),(431356,1451012),(155148,1463028)}

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