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 0cubes in n ways.[1] Such numbers exist for all n, which follows from the analogous result for taxicab numbers.
Only 10 cabtaxi numbers are known (sequence A047696 in the OEIS):
History
Cabtaxi(2) was known to François Viète and Pietro Bongo in the late 16th century in the equivalent form . 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.[1]
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.[1] 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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