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 9 are known :
:

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}

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.

External links

★ Announcement of Cabtaxi(9)

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