Smallest positive integer written as the sum of two integer cubes in n ways
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.
Known cabtaxi 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.
Smallest positive integer written as the sum of two integer cubes in n ways
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.
Known cabtaxi 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.