Is 1729 the only number that is the sum of two cubes in two different ways?

[ Editor: in reply to Eric Ivancich's claim, "Apparently, there is only
one number that has the property of being the sum of two cubes in two
different ways." ]

Please define "Apparently".

 1729 = 12^3 + 1^3 = 10^3 + 9^3
 4104 = 16^3 + 2^3 = 15^3 + 9^3
 13832 = 24^3 + 2^3 = 20^3 + 18^3
 20683 = 27^3 + 10^3 = 24^3 + 19^3
 32832 = 32^3 + 4^3 = 30^3 + 18^3
 39312 = 34^3 + 2^3 = 33^3 + 15^3
 40033 = 34^3 + 9^3 = 33^3 + 16^3
 46683 = 36^3 + 3^3 = 30^3 + 27^3
 64232 = 39^3 + 17^3 = 36^3 + 26^3
 65728 = 40^3 + 12^3 = 33^3 + 31^3
 110656 = 48^3 + 4^3 = 40^3 + 36^3
 110808 = 48^3 + 6^3 = 45^3 + 27^3
 134379 = 51^3 + 12^3 = 43^3 + 38^3
 149389 = 53^3 + 8^3 = 50^3 + 29^3
 165464 = 54^3 + 20^3 = 48^3 + 38^3
 171288 = 55^3 + 17^3 = 54^3 + 24^3
 195841 = 58^3 + 9^3 = 57^3 + 22^3
 216027 = 60^3 + 3^3 = 59^3 + 22^3
 216125 = 60^3 + 5^3 = 50^3 + 45^3
 262656 = 64^3 + 8^3 = 60^3 + 36^3
 314496 = 68^3 + 4^3 = 66^3 + 30^3
 320264 = 68^3 + 18^3 = 66^3 + 32^3
 327763 = 67^3 + 30^3 = 58^3 + 51^3
 373464 = 72^3 + 6^3 = 60^3 + 54^3
 402597 = 69^3 + 42^3 = 61^3 + 56^3
 439101 = 76^3 + 5^3 = 69^3 + 48^3
 443889 = 76^3 + 17^3 = 73^3 + 38^3
 513000 = 80^3 + 10^3 = 75^3 + 45^3
 513856 = 78^3 + 34^3 = 72^3 + 52^3
 515375 = 80^3 + 15^3 = 71^3 + 54^3
 525824 = 80^3 + 24^3 = 66^3 + 62^3
 558441 = 81^3 + 30^3 = 72^3 + 57^3
 593047 = 84^3 + 7^3 = 70^3 + 63^3
 684019 = 82^3 + 51^3 = 75^3 + 64^3
 704977 = 89^3 + 2^3 = 86^3 + 41^3
 805688 = 93^3 + 11^3 = 92^3 + 30^3
 842751 = 94^3 + 23^3 = 84^3 + 63^3
 885248 = 96^3 + 8^3 = 80^3 + 72^3
 886464 = 96^3 + 12^3 = 90^3 + 54^3
 920673 = 97^3 + 20^3 = 96^3 + 33^3
 955016 = 98^3 + 24^3 = 89^3 + 63^3
 984067 = 98^3 + 35^3 = 92^3 + 59^3
 994688 = 99^3 + 29^3 = 92^3 + 60^3
 1009736 = 96^3 + 50^3 = 93^3 + 59^3
 1016496 = 97^3 + 47^3 = 90^3 + 66^3
 1061424 = 102^3 + 6^3 = 99^3 + 45^3
 1073375 = 102^3 + 23^3 = 95^3 + 60^3
 1075032 = 102^3 + 24^3 = 86^3 + 76^3
 1080891 = 102^3 + 27^3 = 99^3 + 48^3
 1092728 = 103^3 + 1^3 = 94^3 + 64^3
 1195112 = 106^3 + 16^3 = 100^3 + 58^3
 1260441 = 108^3 + 9^3 = 90^3 + 81^3
 1323712 = 108^3 + 40^3 = 96^3 + 76^3
 1331064 = 110^3 + 4^3 = 101^3 + 67^3
 1370304 = 110^3 + 34^3 = 108^3 + 48^3
 1407672 = 112^3 + 14^3 = 105^3 + 63^3
 1533357 = 109^3 + 62^3 = 93^3 + 90^3
 1566728 = 116^3 + 18^3 = 114^3 + 44^3
 1609272 = 113^3 + 55^3 = 94^3 + 92^3
 1728216 = 120^3 + 6^3 = 118^3 + 44^3
 1729000 = 120^3 + 10^3 = 100^3 + 90^3
 1734264 = 117^3 + 51^3 = 108^3 + 78^3
 1774656 = 120^3 + 36^3 = 99^3 + 93^3
 1845649 = 121^3 + 42^3 = 120^3 + 49^3
 2048391 = 127^3 + 2^3 = 106^3 + 95^3
 2101248 = 128^3 + 16^3 = 120^3 + 72^3
 2301299 = 132^3 + 11^3 = 110^3 + 99^3
 2418271 = 134^3 + 23^3 = 116^3 + 95^3
 2515968 = 136^3 + 8^3 = 132^3 + 60^3
 2562112 = 136^3 + 36^3 = 132^3 + 64^3
 2585375 = 135^3 + 50^3 = 120^3 + 95^3
 2622104 = 134^3 + 60^3 = 116^3 + 102^3
 2691451 = 139^3 + 18^3 = 123^3 + 94^3
 2864288 = 142^3 + 10^3 = 121^3 + 103^3
 2987712 = 144^3 + 12^3 = 120^3 + 108^3
 2991816 = 144^3 + 18^3 = 135^3 + 81^3
 3220776 = 138^3 + 84^3 = 122^3 + 112^3
 3242197 = 141^3 + 76^3 = 138^3 + 85^3
 3375001 = 150^3 + 1^3 = 144^3 + 73^3
 3375008 = 150^3 + 2^3 = 141^3 + 83^3
 3511872 = 152^3 + 4^3 = 151^3 + 41^3
 3512808 = 152^3 + 10^3 = 138^3 + 96^3
 3551112 = 152^3 + 34^3 = 146^3 + 76^3
 3587409 = 153^3 + 18^3 = 122^3 + 121^3
 3628233 = 153^3 + 36^3 = 129^3 + 114^3
 3798613 = 156^3 + 13^3 = 130^3 + 117^3
 3813992 = 156^3 + 26^3 = 125^3 + 123^3
 4033503 = 159^3 + 24^3 = 150^3 + 87^3
 4104000 = 160^3 + 20^3 = 150^3 + 90^3
 4110848 = 156^3 + 68^3 = 144^3 + 104^3
 4123000 = 160^3 + 30^3 = 142^3 + 108^3
 4174281 = 161^3 + 10^3 = 153^3 + 84^3
 4206592 = 160^3 + 48^3 = 132^3 + 124^3
 4342914 = 163^3 + 23^3 = 137^3 + 121^3
 4467528 = 162^3 + 60^3 = 144^3 + 114^3
 4505949 = 165^3 + 24^3 = 157^3 + 86^3
 4511808 = 165^3 + 27^3 = 152^3 + 100^3
 4607064 = 166^3 + 32^3 = 135^3 + 129^3
 4624776 = 165^3 + 51^3 = 162^3 + 72^3
 4673088 = 167^3 + 25^3 = 164^3 + 64^3
 4744376 = 168^3 + 14^3 = 140^3 + 126^3
 4914000 = 170^3 + 10^3 = 165^3 + 75^3
 4931101 = 165^3 + 76^3 = 157^3 + 102^3
 5004125 = 170^3 + 45^3 = 165^3 + 80^3
 5135104 = 172^3 + 36^3 = 163^3 + 93^3
 5287707 = 174^3 + 27^3 = 171^3 + 66^3
 5318677 = 174^3 + 37^3 = 141^3 + 136^3
 5446359 = 172^3 + 71^3 = 144^3 + 135^3
 5462424 = 176^3 + 22^3 = 165^3 + 99^3
 5472152 = 164^3 + 102^3 = 150^3 + 128^3
 5639816 = 178^3 + 4^3 = 172^3 + 82^3
 5772403 = 178^3 + 51^3 = 162^3 + 115^3
 5799339 = 179^3 + 40^3 = 166^3 + 107^3
 5832729 = 180^3 + 9^3 = 177^3 + 66^3
 5835375 = 180^3 + 15^3 = 150^3 + 135^3
 6013000 = 178^3 + 72^3 = 165^3 + 115^3
 6017193 = 180^3 + 57^3 = 166^3 + 113^3
 6058747 = 172^3 + 99^3 = 163^3 + 120^3
 6230504 = 184^3 + 10^3 = 181^3 + 67^3
 6434883 = 186^3 + 3^3 = 184^3 + 59^3
 6445504 = 186^3 + 22^3 = 184^3 + 60^3
 6517000 = 184^3 + 66^3 = 175^3 + 105^3
 6742008 = 188^3 + 46^3 = 168^3 + 126^3
 7081984 = 192^3 + 16^3 = 160^3 + 144^3
 7091712 = 192^3 + 24^3 = 180^3 + 108^3
 7094269 = 189^3 + 70^3 = 168^3 + 133^3
 7882245 = 192^3 + 93^3 = 166^3 + 149^3
 8077888 = 192^3 + 100^3 = 186^3 + 118^3

-- Matt Crawford <matt@oddjob.uchicago.edu>

[ Editor: every line above checks, but the list is not complete below
its last entry: for example 7365384 = 194^3 + 40^3 = 192^3 + 66^3 is
missing, and so are five other numbers between 7.3 and 8.08 million. ]