Cabtaxi number

Cabtaxi number

In mathematics, the "n"-th cabtaxi number, typically denoted Cabtaxi("n"), is defined as the smallest positive integer that can be written as the sum of two "positive or negative or 0" cubes in "n" ways. Such numbers exist for all "n" (since taxicab numbers exist for all "n"); however, only 10 are known OEIS|id=A047696:

:egin{matrix}mathrm{Cabtaxi}(1)&=&1&=&1^3 pm 0^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(2)&=&91&=&3^3 + 4^3 \&&&=&6^3 - 5^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(3)&=&728&=&6^3 + 8^3 \&&&=&9^3 - 1^3 \&&&=&12^3 - 10^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \&&&=&140^3 - 14^3 \&&&=&168^3 - 126^3 \&&&=&207^3 - 183^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(5)&=&6017193&=&166^3 + 113^3 \&&&=&180^3 + 57^3 \&&&=&185^3 - 68^3 \&&&=&209^3 - 146^3 \&&&=&246^3 - 207^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(6)&=&1412774811&=&963^3 + 804^3 \&&&=&1134^3 - 357^3 \&&&=&1155^3 - 504^3 \&&&=&1246^3 - 805^3 \&&&=&2115^3 - 2004^3 \&&&=&4746^3 - 4725^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(7)&=&11302198488&=&1926^3 + 1608^3 \&&&=&1939^3 + 1589^3 \&&&=&2268^3 - 714^3 \&&&=&2310^3 - 1008^3 \&&&=&2492^3 - 1610^3 \&&&=&4230^3 - 4008^3 \&&&=&9492^3 - 9450^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(8)&=&137513849003496&=&22944^3 + 50058^3 \&&&=&36547^3 + 44597^3 \&&&=&36984^3 + 44298^3 \&&&=&52164^3 - 16422^3 \&&&=&53130^3 - 23184^3 \&&&=&57316^3 - 37030^3 \&&&=&97290^3 - 92184^3 \&&&=&218316^3 - 217350^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(9)&=&424910390480793000&=&645210^3 + 538680^3 \&&&=&649565^3 + 532315^3 \&&&=&752409^3 - 101409^3 \&&&=&759780^3 - 239190^3 \&&&=&773850^3 - 337680^3 \&&&=&834820^3 - 539350^3 \&&&=&1417050^3 - 1342680^3 \&&&=&3179820^3 - 3165750^3 \&&&=&5960010^3 - 5956020^3end{matrix}

:egin{matrix}mathrm{Cabtaxi}(10)&=&933528127886302221000&=&77480130^3 - 77428260^3 \&&&=&41337660^3 - 41154750^3 \&&&=&18421650^3 - 17454840^3 \&&&=&10852660^3 - 7011550^3 \&&&=&10060050^3 - 4389840^3 \&&&=&9877140^3 - 3109470^3 \&&&=&9781317^3 - 1318317^3 \&&&=&9773330^3 - 84560^3 \&&&=&8444345^3 + 6920095^3 \&&&=&8387730^3 + 7002840^3end{matrix}

Cabtaxi(5), Cabtaxi(6) and Cabtaxi(7) were found by Randall L. Rathbun; Cabtaxi(8) was found by Daniel J. Bernstein; Cabtaxi(9) was found by Duncan Moore, using Bernstein's method. Cabtaxi(10) was first reported by Christian Boyer in 2006 and verified as Cabtaxi(10) by Uwe Hollerbach and reported on the NMBRTHRY mailing list on May 16 2008.

See also

* Taxicab number
* Generalized taxicab number

External links

* [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0502&L=nmbrthry&F=&S=&P=55 Announcement of Cabtaxi(9)]
* [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0805&L=nmbrthry&T=0&P=1284 Announcement of Cabtaxi(10)]
* [http://euler.free.fr/ Cabtaxi at Euler]


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