The Taxicab problem

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
For more informations, check http://www.cs.uwaterloo.ca/journals/JIS/wilson10.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

Taxicab(1) = 2
= 1³ + 1³

trivial.

Taxicab(2) = 1729
= 1³ + 12³
= 9³ + 10³

first published by Bernard Frénicle de Bessy in 1657.

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

Cabtaxi(1) = 0
= 1³ - 1³

trivial.

Cabtaxi(2) = 91
= 3³ + 4³
= 6³ - 5³

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(4) = 2741256
= 108³ + 114³
= 140³ - 14³
= 168³ - 126³
= 207³ - 183³

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.