Cabtaxi number
In number theory, 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, which follows from the analogous result for taxicab numbers.
Known cabtaxi numbers
Only 10 cabtaxi numbers are known (sequence A047696 in the OEIS):
[math]\displaystyle{ \begin{align} \mathrm{Cabtaxi}(1) =& \ 1 \\ &= 1^3 + 0^3 \\[6pt] \mathrm{Cabtaxi}(2) =& \ 91 \\ &= 3^3 + 4^3 \\ &= 6^3 - 5^3 \\[6pt] \mathrm{Cabtaxi}(3) =& \ 728 \\ &= 6^3 + 8^3 \\ &= 9^3 - 1^3 \\ &= 12^3 - 10^3 \\[6pt] \mathrm{Cabtaxi}(4) =& \ 2741256 \\ &= 108^3 + 114^3 \\ &= 140^3 - 14^3 \\ &= 168^3 - 126^3 \\ &= 207^3 - 183^3 \\[6pt] \mathrm{Cabtaxi}(5) =& \ 6017193 \\ &= 166^3 + 113^3 \\ &= 180^3 + 57^3 \\ &= 185^3 - 68^3 \\ &= 209^3 - 146^3 \\ &= 246^3 - 207^3 \\[6pt] \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 \\[6pt] \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 \\[6pt] \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 \\[6pt] \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 \\[6pt] \mathrm{Cabtaxi}(10) =& \ 933528127886302221000 \\ &= 8387730^3 + 7002840^3 \\ &= 8444345^3 + 6920095^3 \\ &= 9773330^3 - 84560^3 \\ &= 9781317^3 - 1318317^3 \\ &= 9877140^3 - 3109470^3 \\ &= 10060050^3 - 4389840^3 \\ &= 10852660^3 - 7011550^3 \\ &= 18421650^3 - 17454840^3 \\ &= 41337660^3 - 41154750^3 \\ &= 77480130^3 - 77428260^3 \end{align} }[/math]
History
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.
See also
External links
- Announcement of Cabtaxi(9)
- Announcement of Cabtaxi(10)[yes|permanent dead link|dead link}}]
- Cabtaxi at Euler
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