New Upper Bounds for Taxicab and Cabtaxi Numbers
    Christian Boyer, France, 2006-2011    (Last update: March 24, 2011)


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
(France)

1657

Taxicab(3)

= 87539319

~8.75E+07

John Leech (UK)

1957

Taxicab(4)

= 6963472309248

~6.96E+12

Edwin Rosenstiel, John A. Dardis,
Colin R. Rosenstiel (UK)

1989

Taxicab(5)

= 48988659276962496

~4.90E+16

John A. Dardis (UK)

1994

Taxicab(6)

= 24153319581254312065344

~2.42E+22

Randall L. Rathbun (USA),
Uwe Hollerbach (USA)

July 2002,
March 2008 (*)

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 -
Jaroslaw Wroblewski
(France - Poland)

April 2008

Taxicab(12)

≤ 16119148654034302034428760115512552827992287460693283776000

~1.61E+58

Taxicab(13)
...
Taxicab(22)

see the downloadable file at the end of this web page

~9.88E+64
...
~4.68E+159

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),
Pietro Bongo (Italy) indep.

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),
Uwe Hollerbach (USA)

Dec. 2006,
May 2008 (**)

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
(France - Poland)

April 2008

Cabtaxi(14)

≤ 3424462108508996825708504669331456

~3.42E+33

Duncan Moore (UK)

March 2008

Cabtaxi(15)

≤ 119860206095954108554485737248700928

~1.20E+35

 Christian Boyer -
Jaroslaw Wroblewski
(France - Poland)

 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)
...
Cabtaxi(42)

see the downloadable file at the end of this web page

~7.57E+58
...
~ 9.11E+157

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 taxi-cab 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.”


London Taxi cab n°1729.
Built in France by Unic, they were the most current taxi cabs in London at the time of the Hardy/Ramanujan anecdote.
Image from the "London Vintage Taxi Association"
http://www.lvta.co.uk/history.htm....but slightly modified: LF 5795 becomes here LF 1729!


Pierre de Fermat

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 = 13 + 123 = 93 + 103. 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. 333-334 (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 Boyer-Wroblewski.

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 6-7-way Taxicab solutions up to 5 * 1028 and my list of 9-10-11-way Cabtaxi solutions up to 1.2 * 1025. Who will confirm these lists including Taxicab(7) and Cabtaxi(11)?


2010-2011: Bill Butler on the way to Taxicab(7),
          and Uwe Hollerbach on the way to Cabtaxi(11).


Decompositions of upper bounds

Taxicab(7) ≤  24885189317885898975235988544
(≈ 2.49 * 1028)

1st way

=  26486609663 + 18472821223

2nd way

=  26856356523 + 17667420963

3rd way

=  27364140083 + 16380248683

4th way

=  28944061873 +   8604473813

5th way

=  29157349483 +   4595311283

6th way

=  29183751033 +   3094814733

7th way

=  29195268063 +     587983623

and differences
of cubes

=  49654593643 -  46032446803

=  57025913003 -  54351671363

(...)

Below, no more an upper bound!
This is the real Cabtaxi(10) number.

Cabtaxi(10) = 933528127886302221000
(≈ 9.34 * 1020)

1st way

=   83877303 +  70028403

2nd way

=   84443453 +  69200953

3rd way

=   97733303 -       845603

4th way

=   97813173 -   13183173

5th way

=   98771403 -   31094703

6th way

= 100600503 -   43898403

7th way

= 108526603 -   70115503

8th way

= 184216503 - 174548403

9th way

= 413376603 - 411547503

10th way

= 774801303 - 774282603

Lists of decompositions:

And list of bigger upper bounds: