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)

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.

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?

Definitions:

• Taxicab(n) is the smallest number expressible as a sum of two positive cubes in n different ways
• Cabtaxi(n) is the smallest number expressible as a sum of two positive or negative cubes in n different ways (for example: Cabtaxi(2) = 91 = 3³ + 4³ = 6³ - 5³)

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).

• For more information on Taxicab and Cabtaxi numbers:
• http://euler.free.fr/taxicab.htm (Jean-Charles Meyrignac) and http://homepage.ntlworld.com/duncan-moore/taxicab/index.htm (Duncan Moore)
• http://mathworld.wolfram.com/TaxicabNumber.html and http://mathworld.wolfram.com/CabtaxiNumber.html
• http://en.wikipedia.org/wiki/Taxicab_number and http://en.wikipedia.org/wiki/Cabtaxi_number
• http://oeis.org/A011541 and http://oeis.org/A047696
• Part of problem D1, Richard K. Guy, Unsolved Problems in Number Theory, p. 211-212 (third 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.

• "New Upper Bounds for Taxicab and Cabtaxi Numbers" in Journal of Integer Sequences, Volume 11, 2008, article 08.1.6
  This paper on my Taxicab and Cabtaxi research of 2006-2007 can be downloaded from the JIS website since March 7th 2008, http://www.cs.uwaterloo.ca/journals/JIS/vol11.html, but the figures are not very clear. I recommend this other version with exactly the same text, but cleaner figures:
  
• Download "New Upper Bounds for Taxicab and Cabtaxi Numbers" (PDF file, 1.3Mb)
• "Les nombres Taxicab" in Pour La Science (the French edition of Scientific American), Dossier N°59 Jeux Math', avril-juin 2008, pages 26-28
  This second paper is a shorter version in French.

Interesting news, known after the writing of the two above articles:

• March 2008. As mentioned in these articles, the upper bound found in 2002 by Randall L. Rathbun was probably Taxicab(6). It is now sure, his upper bound is really Taxicab(6). Look at the annoucement done by Uwe Hollerbach, USA, http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0803&L=nmbrthry&T=0&P=1059 and his webpage http://www.korgwal.com/ramanujan.
• March-April 2008. In March, Duncan Moore, UK, announced better Cabtaxi(11..15) upper bounds than my published bounds. And in April, with Jaroslaw Wroblewski, Poland, we found better Cabtaxi(13) and Cabtaxi(≥15) bounds than Moore's. We also found better Taxicab(≥11) upper bounds than my published bounds. See above the updated tables of Taxicab and Cabtaxi numbers!
• May 2008. Cabtaxi(10) = 933528127886302221000. My upper bound of Cabtaxi(10) found in Dec 2006 and published in the above articles is now proved to be the real number! Uwe Hollerbach confirmed it by an exhaustive search. Look at his announcement at http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0805&L=nmbrthry&T=0&P=1284. He also confirmed the part < 9.5 * 10²⁰ of my list of 9-10-way and list of 8-9-10-way Cabtaxi solutions, no number is missing.
• April-May 2008. Other interesting results found by Uwe Hollerbach during his search on Cabtaxi(10). On one of the problems presented in Part 8.2 of the JIS paper, he found several 3-way cabtaxi solutions using 4 primes (different from the only known solution), and the first known solutions using 5 primes. This is also the solution of the question 2, Puzzle 386 of Carlos Rivera http://www.primepuzzles.net/puzzles/puzz_386.htm. But the problem of a 3-way solution using 6 primes is still unsolved! After Uwe's search, we know that such a solution (if existing) is bigger than 9.5 * 10²⁰. Who will solve this difficult problem?
• 68913 = 40³ + 17³ = 41³ - 2³ = 89³ - 86³    (previously only known solution using 4 primes)
• 439926932718 = 7639³ - 1801³ = 9007³ - 6625³ = 11257³ - 9955³    (2nd smallest solution using 4 primes, found by Uwe H.)
• 3607745483160 = 15349³ - 2029³ = 18313³ - 13633³ = 29334³ - 27864³    (3rd smallest solution using 4 primes, found by Uwe H.)
• 7070014626807314 = 159535³ + 144379³ = 198817³ - 92399³ =  228757³ - 169859³    (smallest solution using 5 primes, found by Uwe H., March 29, 2008)
• 104886396577678268754 = 4685887³ + 1259051³ = 4873763³ - 2216057³ = 5857309³- 4579915³    (2nd smallest solution using 5 primes, found by Uwe H., April 3, 2008)
• 319969457707558764294 = 7054657³ - 3145699³ = 10174807³ - 9018049³ = 18819961³ - 18513883³    (3rd smallest solution using 5 primes, found by Uwe H., May 11, 2008)
• 771993128736890238906 = 8160617³ + 6113857³ = 8943911³ + 3838075³ = 23118287³ - 22626413³    (4th smallest solution using 5 primes, found by Uwe H., May 12, 2008)
• 1289090709154791499117014 = 102944713³ + 58296773³ = 108883871³ - 12172313³ = 119375213³ - 74413727³    (another solution using 5 primes, found by Jaroslaw W., May 8, 2008)
   
• July 2008. Cabtaxi(10) and my list of 9-way solutions < Cabtaxi(10) also confirmed by Bill Butler, USA, using a Skulltrail computer: http://www.durangobill.com/Ramanujan.html. Bill worked on Cabtaxi(10) since the beginning of April 2008.
• August 2008. Uwe Hollerbach has completed his computation up to 1.5 * 10²¹ : no solution using 6 primes, and no new solution using 5 primes. And we know now that Cabtaxi(11) > 1.5 * 10²¹.
• November 2008. Uwe Hollerbach has reached 2 * 10²¹ : we know now that Cabtaxi(11) > 2 * 10²¹. And using his specific prime scan, up to 2 * 10²² (faster than his main scan) : still no solution using 6 primes, and still no new solution using 5 primes.

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).

• March 2010. Bill Butler works on Taxicab(7) and has reached 10²³ : with his exhaustive search, we know now that Taxicab(7) > 10²³. There are 867 four-way-or-higher solutions < 10²³, but no seven-way.
• April 2010. Uwe Hollerbach works on Cabtaxi(11) and has reached 10²² : with his exhaustive search, we know now that Cabtaxi(11) > 10²². There are 3535 six-way, 836 seven-way, 170 eight-way, 42 nine-way and 6 ten-way Cabtaxi solutions < 10²², but no eleven-way.
• March 2011. Uwe Hollerbach has reached 2.61858 * 10²², ie exactly 10% of the Cabtaxi(11) upper-bound target. He found 4246 six-way, 997 seven-way, 222 eight-way, 53 nine-way and 8 ten-way Cabtaxi solutions < 2.61858 * 10²², but still no eleven-way, still no solution using 6 primes, and still no new solution using 5 primes. The only known solutions using 5 primes remain as listed above.
  My above list of 9-10-11-way Cabtaxi solutions computed in 2008 seems correct... at least at this point of the search... the 53 nine-way and 8 ten-way solutions found by Uwe being exactly the same.

Decompositions of upper bounds

(...)

Lists of decompositions:

• TaxicabUpdated.txt   Taxicab(7..12)
• CabtaxiUpdated.txt   Cabtaxi(10..22)

And list of bigger upper bounds:

• TaxibigUpdated.txt   Taxicab(13..22) and Cabtaxi (23..42)