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| 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 number]]s exist for all ''n''); however, only 10 are known {{OEIS|id=A047696}}:
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| :<math>\begin{matrix}\mathrm{Cabtaxi}(1)&=&1&=&1^3 \pm 0^3\end{matrix}</math> | | Also visit my web site :: [http://youulike.com/blogs/77778/435421/new-article-reveals-the-low-down Nya online casinon] |
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| :<math>\begin{matrix}\mathrm{Cabtaxi}(2)&=&91&=&3^3 + 4^3 \\&&&=&6^3 - 5^3\end{matrix}</math> | |
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| :<math>\begin{matrix}\mathrm{Cabtaxi}(3)&=&728&=&6^3 + 8^3 \\&&&=&9^3 - 1^3 \\&&&=&12^3 - 10^3\end{matrix}</math> | |
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| :<math>\begin{matrix}\mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \\&&&=&140^3 - 14^3 \\&&&=&168^3 - 126^3 \\&&&=&207^3 - 183^3\end{matrix}</math>
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| :<math>\begin{matrix}\mathrm{Cabtaxi}(5)&=&6017193&=&166^3 + 113^3 \\&&&=&180^3 + 57^3 \\&&&=&185^3 - 68^3 \\&&&=&209^3 - 146^3 \\&&&=&246^3 - 207^3\end{matrix}</math>
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| :<math>\begin{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 - 4725^3\end{matrix}</math>
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| :<math>\begin{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 - 9450^3\end{matrix}</math>
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| :<math>\begin{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 - 217350^3\end{matrix}</math>
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| :<math>\begin{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 - 5956020^3\end{matrix}</math>
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| :<math>\begin{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 + 7002840^3\end{matrix}</math>
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| 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 as an upper bound by [[Christian Boyer]] in 2006 and verified as Cabtaxi(10) by [[Uwe Hollerbach]] and reported on the [[NMBRTHRY]] mailing list on May 16, 2008.
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| == See also==
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| * [[Taxicab number]]
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| * [[Generalized taxicab number]]
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| == External links ==
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| * [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0502&L=nmbrthry&F=&S=&P=55 Announcement of Cabtaxi(9)]
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| * [http://www.korgwal.com/ramanujan/announce_ct10.html Announcement of Cabtaxi(10)]
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| * [http://euler.free.fr/ Cabtaxi at Euler]
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| [[Category:Number theory]]
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