Cabtaxi number
In number theory, the Template:Mvar-th cabtaxi number, typically denoted Template:Math, is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in Template:Mvar ways.Template:R Such numbers exist for all Template:Mvar, which follows from the analogous result for taxicab numbers.
Known cabtaxi numbersEdit
Only 10 cabtaxi numbers are known (sequence A047696 in the OEIS):
<math display=block>\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>
HistoryEdit
Cabtaxi(2) was known to François Viète and Pietro Bongo in the late 16th century in the equivalent form <math>3^3+4^3+5^3=6^3</math>. The existence of Cabtaxi(3) was known to Leonhard Euler, but its actual solution was not found until later, by Edward B. Escott in 1902.Template:R
Cabtaxi(4) through and Cabtaxi(7) were found by Randall L. Rathbun in 1992; Cabtaxi(8) was found by Daniel J. Bernstein in 1998. Cabtaxi(9) was found by Duncan Moore in 2005, using Bernstein's method.Template:R 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.