New Upper Bounds for Taxicab and Cabtaxi Numbers 
Updated tables of best known results
(see also the previous
tables of the best known results, when the JIS and PLS papers were
written in 2007)
Taxicab(2) 
= 1729 
~1.73E+03 
Bernard Frenicle de Bessy 
1657 
Taxicab(3) 
= 87539319 
~8.75E+07 
John Leech (UK) 
1957 
Taxicab(4) 
= 6963472309248 
~6.96E+12 
Edwin Rosenstiel, John A. Dardis, 
1989 
Taxicab(5) 
= 48988659276962496 
~4.90E+16 
John A. Dardis (UK) 
1994 
Taxicab(6) 
= 24153319581254312065344 
~2.42E+22 
Randall L. Rathbun (USA), 
July 2002, 
Taxicab(7) 
≤ 24885189317885898975235988544 
~2.49E+28 
Christian Boyer (France) 
Dec. 2006 
Taxicab(8) 
≤ 50974398750539071400590819921724352 
~5.10E+34 

Taxicab(9) 
≤ 136897813798023990395783317207361432493888 
~1.37E+41 

Taxicab(10) 
≤ 7335345315241855602572782233444632535674275447104 
~7.34E+48 

Taxicab(11) 
≤ 87039729655193781808322993393446581825405320183232000 
~8.70E+52 
Christian Boyer  
April 2008 
Taxicab(12) 
≤ 16119148654034302034428760115512552827992287460693283776000 
~1.61E+58 

Taxicab(13) 
see the downloadable file at the end of this web page 
~9.88E+64 
In the second column, numbers
with background are best known upper bounds: not sure that
they are the true Taxicab numbers, but they may have a chance.
(*) Randall L. Rathbun found this number
as an upper bound. Six years later, this number is proved to be Taxicab(6)
by Uwe Hollerbach, USA.
Cabtaxi(2) 
= 91 
=9.1E+01 
François Viète (France), 
1591 
Cabtaxi(3) 
= 728 
=7.28E+02 
Edward B. Escott (USA) 
1902 
Cabtaxi(4) 
= 2741256 
~2.74E+06 
Randall L. Rathbun (USA) 
~1992 
Cabtaxi(5) 
= 6017193 
~6.02E+06 

Cabtaxi(6) 
= 1412774811 
~1.41E+09 

Cabtaxi(7) 
= 11302198488 
~1.13E+10 

Cabtaxi(8) 
= 137513849003496 
~1.38E+14 
Daniel. J. Bernstein (USA) 
1998 
Cabtaxi(9) 
= 424910390480793000 
~4.25E+17 
Duncan Moore (UK) 
Feb. 2005 
Cabtaxi(10) 
= 933528127886302221000 
~9.34E+20 
Christian Boyer (France), 
Dec. 2006, 
Cabtaxi(11) 
≤ 261858398098545372249216 
~2.62E+23 
Duncan Moore (UK) 
March 2008 
Cabtaxi(12) 
≤ 1796086752557922708257372544 
~1.80E+27 

Cabtaxi(13) 
≤ 308110458144384714689809795584 
~3.08E+29 
C. Boyer  J. Wroblewski 
April 2008 
Cabtaxi(14) 
≤ 3424462108508996825708504669331456 
~3.42E+33 
Duncan Moore (UK) 
March 2008 
Cabtaxi(15) 
≤ 119860206095954108554485737248700928 
~1.20E+35 
Christian Boyer  
April 2008 
Cabtaxi(16) 
≤ 822121153612149230575217671788839665152 
~8.22E+38 

Cabtaxi(17) 
≤ 228528345587492406268587814296067158147072 
~2.29E+41 

Cabtaxi(18) 
≤ 1567475922384610414596243818256724637730766848 
~1.57E+45 

Cabtaxi(19) 
≤ 22388474568951577754900099772066812785435844544000 
~2.24E+49 

Cabtaxi(20) 
≤ 3901860835762247103510236821129665273758992896000000 
~3.90E+51 

Cabtaxi(21) 
≤ 1494725379426214299719362865171579535464276835200448000 
~1.49E+54 

Cabtaxi(22) 
≤ 2804160172002816034210551378130963637402667204941047872000 
~2.80E+57 

Cabtaxi(23) 
see the downloadable file at the end of this web page 
~7.57E+58 
In the second column, numbers
with background are best known upper bounds: not sure that
they are the true Cabtaxi numbers, but they may have a chance.
(**) I found this number
as an upper bound. One year and a half later, this number is proved to be Cabtaxi(10)
by Uwe Hollerbach, USA. And also confirmed by Bill Butler, USA,
in July 2008
What is a "Taxicab" or "Cabtaxi" number?
Famous anecdote on Ramanujan related by G.H. Hardy: I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxicab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.” 


But this problem was much older. 350 years ago, in 1657, Pierre de Fermat asked the question: Trouver deux nombres cubes dont la somme soit égale à deux autres nombres cubes Bernard Frenicle de Bessy found several solutions, including the number 1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}. This is the smallest solution to Fermat's problem. Today, this number is called Taxicab(2), in memory of the Hardy/Ramanujan anecdote. 
Definitions:
Fermat proved that numbers expressible as a sum of two cubes in n different ways exist for any n: see the Theorem 412 of Hardy & Wright, An Introduction of Theory of Numbers, p. 333334 (fifth edition).
2008: Publication of my JIS and PLS papers,
Taxicab(6) and
Cabtaxi(10) now proved by Uwe Hollerbach,
better bounds
by Moore and by BoyerWroblewski.
Interesting news, known after the writing of the two above articles:
The next steps are Taxicab(7) and Cabtaxi(11): are upper bounds in the above tables the real numbers? Here are my list of 67way Taxicab solutions up to 5 * 10^{28} and my list of 91011way Cabtaxi solutions up to 1.2 * 10^{25}. Who will confirm these lists including Taxicab(7) and Cabtaxi(11)?
20102011: Bill Butler on the way
to Taxicab(7),
and
Uwe Hollerbach on the way to Cabtaxi(11).
Decompositions of upper bounds
Taxicab(7) ≤ 24885189317885898975235988544 

1st way 
= 2648660966^{3} + 1847282122^{3} 
2nd way 
= 2685635652^{3} + 1766742096^{3} 
3rd way 
= 2736414008^{3} + 1638024868^{3} 
4th way 
= 2894406187^{3} + 860447381^{3} 
5th way 
= 2915734948^{3} + 459531128^{3} 
6th way 
= 2918375103^{3} + 309481473^{3} 
7th way 
= 2919526806^{3} + 58798362^{3} 
and differences 
= 4965459364^{3}  4603244680^{3} 
= 5702591300^{3}  5435167136^{3} 
(...)
And list of bigger upper bounds: