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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72" style='word-wrap:break-word'><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Thx Giovanni,<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Do allow me to propose some terminology. Any number which can be expressed as the sum of two cubes two or more different ways is a taxicab number.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Any taxicab number which can be expressed as sum of cubes exactly 2 ways is a 2 taxi or double taxi as you suggested. All of those on your list below are double taxis.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Any number which can be expressed as the sum of cubes exactly three ways is a triple taxi and so on.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>I found 8 triple taxis yesterday and will see if I can find about the same number in the next half hour. Hint: I discovered a marvelous algorithm, not the one posted earlier. I think I shall call it… hmmm… something creative, such as... spike’s taxi algorithm.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>spike<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><div style='border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0in 0in 0in'><p class=MsoNormal><b>From:</b> extropy-chat <extropy-chat-bounces@lists.extropy.org> <b>On Behalf Of </b>Giovanni Santostasi via extropy-chat<br><b>Sent:</b> Monday, 23 January, 2023 7:59 PM<br><b>To:</b> ExI chat list <extropy-chat@lists.extropy.org><br><b>Cc:</b> Giovanni Santostasi <gsantostasi@gmail.com><br><b>Subject:</b> Re: [ExI] taxi numbers<o:p></o:p></p></div><p class=MsoNormal><o:p> </o:p></p><div><div><p class=MsoNormal>Spike,<o:p></o:p></p><div><p class=MsoNormal style='margin-bottom:12.0pt'>Here the 2 Taxi numbers up to 10^7. I made also a map of the components (first number bigger in green and second number red). Not sure it means anything but there is an interesting gap in the middle. I will try the 3 Taxi numbers overnight. In the following list the first number is the taxi number, the following 2 are the cubes of the first sum and the 2 following numbers are the cubes of the second sum. <br><br>1729 12 1 9 10<br>4104 16 2 9 15<br>13832 24 2 18 20<br>20683 27 10 19 24<br>32832 32 4 18 30<br>39312 34 2 15 33<br>40033 34 9 16 33<br>46683 36 3 27 30<br>64232 39 17 26 36<br>65728 40 12 31 33<br>110656 48 4 36 40<br>110808 48 6 27 45<br>134379 51 12 38 43<br>149389 53 8 29 50<br>165464 54 20 38 48<br>171288 55 17 24 54<br>195841 58 9 22 57<br>216027 60 3 22 59<br>216125 60 5 45 50<br>262656 64 8 36 60<br>314496 68 4 30 66<br>320264 68 18 32 66<br>327763 67 30 51 58<br>373464 72 6 54 60<br>402597 69 42 56 61<br>439101 76 5 48 69<br>443889 76 17 38 73<br>513000 80 10 45 75<br>513856 78 34 52 72<br>515375 80 15 54 71<br>525824 80 24 62 66<br>558441 81 30 57 72<br>593047 84 7 63 70<br>684019 82 51 64 75<br>704977 89 2 41 86<br>805688 93 11 30 92<br>842751 94 23 63 84<br>885248 96 8 72 80<br>886464 96 12 54 90<br>920673 97 20 33 96<br>955016 98 24 63 89<br>984067 98 35 59 92<br>994688 99 29 60 92<br>1009736 96 50 59 93<br>1016496 97 47 66 90<br>1061424 102 6 45 99<br>1073375 102 23 60 95<br>1075032 102 24 76 86<br>1080891 102 27 48 99<br>1092728 103 1 64 94<br>1195112 106 16 58 100<br>1260441 108 9 81 90<br>1323712 108 40 76 96<br>1331064 110 4 67 101<br>1370304 110 34 48 108<br>1407672 112 14 63 105<br>1533357 109 62 90 93<br>1566728 116 18 44 114<br>1609272 113 55 92 94<br>1728216 120 6 44 118<br>1729000 120 10 90 100<br>1734264 117 51 78 108<br>1774656 120 36 93 99<br>1845649 121 42 49 120<br>2048391 127 2 95 106<br>2101248 128 16 72 120<br>2301299 132 11 99 110<br>2418271 134 23 95 116<br>2515968 136 8 60 132<br>2562112 136 36 64 132<br>2585375 135 50 95 120<br>2622104 134 60 102 116<br>2691451 139 18 94 123<br>2864288 142 10 103 121<br>2987712 144 12 108 120<br>2991816 144 18 81 135<br>3220776 138 84 112 122<br>3242197 141 76 85 138<br>3375001 150 1 73 144<br>3375008 150 2 83 141<br>3511872 152 4 41 151<br>3512808 152 10 96 138<br>3551112 152 34 76 146<br>3587409 153 18 121 122<br>3628233 153 36 114 129<br>3798613 156 13 117 130<br>3813992 156 26 123 125<br>4033503 159 24 87 150<br>4104000 160 20 90 150<br>4110848 156 68 104 144<br>4123000 160 30 108 142<br>4174281 161 10 84 153<br>4206592 160 48 124 132<br>4342914 163 23 121 137<br>4467528 162 60 114 144<br>4505949 165 24 86 157<br>4511808 165 27 100 152<br>4607064 166 32 129 135<br>4624776 165 51 72 162<br>4673088 167 25 64 164<br>4744376 168 14 126 140<br>4914000 170 10 75 165<br>4931101 165 76 102 157<br>5004125 170 45 80 165<br>5135104 172 36 93 163<br>5287707 174 27 66 171<br>5318677 174 37 136 141<br>5446359 172 71 135 144<br>5462424 176 22 99 165<br>5472152 164 102 128 150<br>5639816 178 4 82 172<br>5772403 178 51 115 162<br>5799339 179 40 107 166<br>5832729 180 9 66 177<br>5835375 180 15 135 150<br>6013000 178 72 115 165<br>6017193 180 57 113 166<br>6058747 172 99 120 163<br>6230504 184 10 67 181<br>6434883 186 3 59 184<br>6445504 186 22 60 184<br>6517000 184 66 105 175<br>6742008 188 46 126 168<br>7081984 192 16 144 160<br>7091712 192 24 108 180<br>7094269 189 70 133 168<br>7365384 194 40 66 192<br>7620661 196 45 133 174<br>7640128 196 48 126 178<br>7872536 196 70 118 184<br>7882245 192 93 149 166<br>7957504 198 58 120 184<br>8029000 195 85 130 180<br>8077888 192 100 118 186<br>8131968 194 94 132 180<br>8216000 200 60 155 165<br>8275176 202 32 129 183<br>8491392 204 12 90 198<br>8493039 204 15 79 200<br>8494577 204 17 153 170<br>8587000 204 46 120 190<br>8600256 204 48 152 172<br>8647128 204 54 96 198<br>8741824 206 2 128 188<br>8849601 201 90 153 174<br>8872487 207 14 140 183<br>9016488 208 26 117 195<br>9261729 210 9 161 172<br>9443761 209 68 146 185<br>9560896 212 32 116 200<br><br><o:p></o:p></p></div></div></div></div></body></html>