In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the -th cabtaxi number, typically denoted , is defined as the smallest positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
that can be written as the sum of two ''positive or negative or 0''
cubes
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
in ways. Such numbers exist for all , which follows from the analogous result for
taxicab numbers.
Known cabtaxi numbers
Only 10 cabtaxi numbers are known :
History
Cabtaxi(2) was known to
François Viète
François Viète (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French people, French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as par ...
and
Pietro Bongo in the late 16th century in the equivalent form
. The existence of Cabtaxi(3) was known to
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, but its actual solution was not found until later, by Edward B. Escott in 1902.
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. 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
References
{{reflist, refs=
[{{citation
, last = Boyer , first = Christian
, issue = 1
, journal = Journal of Integer Sequences
, mr = 2391298
, article-number = 08.1.6
, title = New upper bounds for taxicab and cabtaxi numbers
, url = https://cs.uwaterloo.ca/journals/JIS/VOL11/Boyer/boyer.pdf
, volume = 11
, year = 2008]
External links
Announcement of Cabtaxi(9)Announcement of Cabtaxi(10)Cabtaxi at Euler
Number theory