THE TAXICAB PROBLEM

To many mathematicians, the mere mention of the number 1729 recalls the following incident involving mathematicians G.H. Hardy and Srinivasa Ramanujan:

Once, in the taxi from London [to Putney], Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen.

"No, Hardy," said Ramanujan. "It is a very interesting number. It is the smallest number expressible as the sum of two [positive] cubes in two different ways."

In memory of this incident, the least number which is the sum of two positive cubes in n different ways is called the nth taxicab number, which I will denote Taxicab(n). It is shown that for any n >= 1, there indeed exist numbers which are the sum of two positive cubes in n ways, which guarantees the existence of Taxicab(n) for n >= 1.
Let's note Cabtaxi(n) the smallest positive integer which can be written as the sum of two positive or negative cubes in n ways.

Extract from http://pi.lacim.uqam.ca/eng/problem_en.html
A more recent article is available here: http://www.sciencenews.org/20020727/mathtrek.asp
Rosenstiel's site is here: http://www.cix.co.uk/~rosenstiel/cubes/welcome.htm
The sequences below are A011541 and A047696 sequences of the On-Line Encyclopedia of Integer Sequences.

For the most recent work, check Christian Boyer's page: http://cboyer.club.fr/Taxicab.htm

trivial.

## Taxicab(3) = 87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143

found by Leech in 1957.

## Taxicab(4) = 6963472309248 = 24213 + 190833 = 54363 + 189483 = 102003 + 180723 = 133223 + 166303

found by E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel in 1991.

## Taxicab(5) = 48988659276962496 = 387873 + 3657573 = 1078393 + 3627533 = 2052923 + 3429523 = 2214243 + 3365883 = 2315183 + 3319543

found by David Wilson on November 21, 1997 then by D.J Bernstein in 1998, and by Tomas Womack in December 1998.
Other solutions found by David Wilson are 490593422681271000, 6355491080314102272, 27365551142421413376, 1199962860219870469632, 111549833098123426841016.
D.J Bernstein discovered 391909274215699968 in 1998.

## Taxicab(6) <= 24153319581254312065344 = 289062063 + 5821623 = 288948033 + 30641733 = 286574873 + 85192813 = 270932083 + 162180683 = 265904523 + 174924963 = 262243663 + 182899223

found by Randall L. Rathbun in 2002
David Wilson discovered 8230545258248091551205888 in 1997.
D.J Bernstein verified that Taxicab(6)>= 1018
Stuart Gascoigne verified that Taxicab(6) > 6.8*10^19 in May 2003.
C.S Calude, E. Calude and M.J Dinneen showed that 24153319581254312065344 was Taxicab(6) with a probability of 99%

## Taxicab(7) <= 24885189317885898975235988544 = 26486609663 + 18472821223 = 26856356523 + 17667420963 = 27364140083 + 16380248683 = 28944061873 + 8604473813 = 29157349483 + 4595311283 = 29183751033 + 3094814733 = 29195268063 + 587983623

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

## Taxicab(8) <= 50974398750539071400590819921724352 = 2995120635763 + 2888736628763 = 3363799426823 + 2346048294943 = 3410757278043 + 2243762461923 = 3475245790163 + 2080291582363 = 3675895857493 + 1092768173873 = 3702983383963 + 583604532563 = 3706336380813 + 393041470713 = 3707799043623 + 74673919743

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

## Taxicab(9) <= 136897813798023990395783317207361432493888 = 416321768370643 + 401534391397643 = 467568120327983 + 326100712996663 = 474095261647563 + 311882982206883 = 483059164832243 + 289160529948043 = 510949524191113 + 151894776167933 = 514714690370443 + 81121030025843 = 515180756932593 + 54632764428693 = 515300421426563 + 40768778055883 = 515384067063183 + 10379674843863

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

## Taxicab(10) <= 7335345315241855602572782233444632535674275447104 = 156953306675731283 + 151378465556910283 = 176273181363648463 + 122939968799740823 = 178733913641130123 + 117579884291993763 = 182113305141754483 + 109013519790411083 = 192627970620048473 + 57264330615309613 = 194047438269655883 + 30582628319741683 = 194223145363586433 + 20596552189616133 = 194268258877813123 + 15369829327066763 = 194293797782705603 + 9040693335688843 = 194299793282818863 + 3913137416135223

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

## Taxicab(11) <= 2818537360434849382734382145310807703728251895897826621632 = 114105053953256640563 + 110052144459873773563 = 128150602851372430423 + 89377357317411576143 = 129939555217101597243 + 85480575880279463523 = 132396372838055506963 + 79252828887628855163 = 136001929743147327863 + 67163799217793993263 = 140040534640775237693 + 41631168357330086473 = 141072487622039824763 + 22233570788452201363 = 141200226679327334613 + 14973693441850926513 = 141233024204170138243 + 11173865920777534523 = 141251590988026971203 + 6572584055045786683 = 141255949716609311223 + 2844850901530304943

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

## Taxicab(12) <= 73914858746493893996583617733225161086864012865017882136931801625152 = 339006115295125479103763 + 326964921190284981246763 = 380735441071427490777823 + 265540128590029792711943 = 386050418550008845400043 + 253962790940310286117923 = 393349623701862911178163 + 235460154625145328680363 = 404061733266890711072063 + 199543647476065953975463 = 416060428417743231176993 + 123686201189627686902373 = 419126360725080319361963 + 66055938812491490240563 = 419505873464281511126313 + 44486843215739102661213 = 419603314910589480711043 + 33197555650630055058923 = 419658476825428131435203 + 19527147227541032226283 = 419658897311362294765263 + 19330975426181222410263 = 419671426608046263634623 + 8452052028446535976743

found by Christian Boyer in 2006
http://cboyer.club.fr/Taxicab.htm

trivial.

## Cabtaxi(3) = 728 = 63 + 83 = 93 - 13 = 123 - 103

Note that the terms 91 and 728 also are numerators of expansion of (1-x)^(-1/3).
Is this a coincidence ?
(This is sequence A004117 from the Online Encyclopedia of Integer Sequences)

## Cabtaxi(5) = 6017193 = 1663 + 1133 = 1803 + 573 = 1853 - 683 = 2093 - 1463 = 2463 - 2073

found by Randall Rathbun.

## Cabtaxi(6) = 1412774811 = 9633 + 8043 = 11343 - 3573 = 11553 - 5043 = 12463 - 8053 = 21153 - 20043 = 47463 - 47253

found by Randall Rathbun.

## Cabtaxi(7) = 11302198488 = 19263 + 16083 = 19393 + 15893 = 22683 - 7143 = 23103 - 10083 = 24923 - 16103 = 42303 - 40083 = 94923 - 94503

found by Randall Rathbun.

## Cabtaxi(8) = 137513849003496 = 229443 + 500583 = 365473 + 445973 = 369843 + 442983 = 521643 - 164223 = 531303 - 231843 = 573163 - 370303 = 972903 - 921843 = 2183163 - 2173503

found by D.J. Bernstein.

## Cabtaxi(9) = 424910390480793000 = 6452103 + 5386803 = 6495653 + 5323153 = 7524093 - 1014093 = 7597803 - 2391903 = 7738503 - 3376803 = 8348203 - 5393503 = 14170503 - 13426803 = 31798203 - 31657503 = 59600103 - 59560203

found by Duncan Moore on the 01/31/2005.
Jaroslaw Wroblewski found 825001442051661504 on the 01/24/2005.
Randall Rathbun found 10933313592720956472 in July 2002.

## Cabtaxi(10) <= 933528127886302221000 = 83877303 + 70028403 = 84443453 + 69200953 = 97733303 - 845603 = 97813173 - 13183173 = 98771403 - 31094703 = 100600503 - 43898403 = 108526603 - 70115503 = 184216503 - 174548403 = 413376603 - 411547503 = 774801303 - 774282603

found by Christian Boyer in 2006.

## Cabtaxi(11) <= 11358236731992639122907000

found by Christian Boyer in 2006.

## Cabtaxi(12) <= 1912223147184127402358643000

found by Christian Boyer in 2006.

## Cabtaxi(13) <= 23266019031789278104497609381000

found by Christian Boyer in 2006.

## Cabtaxi(14) <= 567434938166308703690592195193209000

found by Christian Boyer in 2006.

## Cabtaxi(15) <= 31136289927061691188910174934641764248000

found by Christian Boyer in 2006.

## Cabtaxi(16) <= 1577146493675455843791867090964409284453944000

found by Christian Boyer in 2006.

## Cabtaxi(17) <= 125394186272654467772359976801307288979079725608000

found by Christian Boyer in 2006.

## Cabtaxi(18) <= 768476640447541361234827077147113303513533370352195096000

found by Christian Boyer in 2006.

## Cabtaxi(19) <= 298950477236981197723488725070538575992924211134299879660632000

found by Christian Boyer in 2006.

## Cabtaxi(20) <= 2149172021033860338362430683389430843511963750524516489973424104024000

found by Christian Boyer in 2006.

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