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 numbers

Only 10 cabtaxi numbers are known Template:OEIS:

$$\begin{array}{cc}\mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(1)=& 1\\ & =1^3+0^3\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(2)=& 91\\ & =3^3+4^3\\ & =6^3-5^3\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(3)=& 728\\ & =6^3+8^3\\ & =9^3-1^3\\ & =12^3-10^3\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(4)=& 2741256\\ & =108^3+114^3\\ & =140^3-14^3\\ & =168^3-126^3\\ & =207^3-183^3\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(5)=& 6017193\\ & =166^3+113^3\\ & =180^3+57^3\\ & =185^3-68^3\\ & =209^3-146^3\\ & =246^3-207^3\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(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\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(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\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(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\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(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\\ \mathrm{C}\mathrm{a}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i}(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{array}$$

History

Cabtaxi(2) was known to François Viète and Pietro Bongo in the late 16th century in the equivalent form $3^3+4^3+5^3=6^3$. 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.

See also

• Taxicab number
• Generalized taxicab number

External links

• Announcement of Cabtaxi(9)
• Announcement of Cabtaxi(10)
• Cabtaxi at Euler