OFFSET
1,1
COMMENTS
The sequence is infinite: Fermat proved that numbers expressible as a sum of two positive integral cubes in n different ways exist for any n. Hardy and Wright give a proof in Theorem 412 of An Introduction of Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition).
A001235 gives another definition of "taxicab numbers".
David W. Wilson reports a(6) <= 8230545258248091551205888. [But see next line!]
Randall L Rathbun has shown that a(6) <= 24153319581254312065344.
C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, 2003, show that with high probability, a(6) = 24153319581254312065344.
When negative cubes are allowed, such terms are called "Cabtaxi" numbers, cf. Boyer's web page, Wikipedia or MathWorld. - M. F. Hasler, Feb 05 2013
a(7) <= 24885189317885898975235988544. - Robert G. Wilson v, Nov 18 2012
a(8) <= 50974398750539071400590819921724352 = 58360453256^3 + 370298338396^3 = 7467391974^3 + 370779904362^3 = 39304147071^3 + 370633638081^3 = 109276817387^3 + 367589585749^3 = 208029158236^3 + 347524579016^3 = 224376246192^3 + 341075727804^3 = 234604829494^3 + 336379942682^3 = 288873662876^3 + 299512063576^3. - PoChi Su, May 16 2013
a(9) <= 136897813798023990395783317207361432493888. - PoChi Su, May 17 2013
From PoChi Su, Oct 09 2014: (Start)
The preceding bounds are not the best that are presently known.
An upper bound for a(22) was given by C. Boyer (see the C. Boyer link), namely
BTa(22)= 2^12 *3^9 * 5^9 *7^4 *11^3 *13^6 *17^3 *19^3 *31^4 *37^4 *43 *61^3 *73 *79^3 *97^3 *103^3 *109^3 *127^3 *139^3 *157 *181^3 *197^3 *397^3 *457^3 *503^3 *521^3 *607^3 *4261^3.
We also know that (97*491)^3*BTa(22) is an upper bound on a(23), corresponding to the sum x^3+y^3 with
x=2^5 *3^4 *5^3 *7 *11 *13^2 *17 *19^2 *31 *37 *61 *79 *103 *109 *127 *139 *181 *197 *397 *457 *503 *521 *607 *4261 *11836681,
y=2^4 *3^3 *5^3 *7 *11 *13^2 *17 *19 *31 *37 *61 *79 *89 *103 *109 *127 *139 *181 *197 *397 * 457 *503 * 521 *607 *4261 *81929041.
(End)
Conjecture: the number of distinct prime factors of a(n) is strictly increasing as n grows (this is not true if a(7) is equal to the upper bound given above), but never exceeds 2*n. - Sergey Pavlov, Mar 01 2017
REFERENCES
C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
R. K. Guy, Unsolved Problems in Number Theory, D1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition), see Theorem 412.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.
LINKS
D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
C. Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6.
C. S. & E. Calude and M. T. Dinneen, What is the value of Taxicab(6)?
C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, J. Universal Computer Science, 9 (2003), 1196-1203.
U. Hollerbach, The sixth taxicab number is 24153319581254312065344, posting to the NMBRTHRY mailing list, Mar 09 2008.
Bernd C. Kellner, On primary Carmichael numbers, arXiv:1902.11283 [math.NT], 2019. See also Integers (2022) Vol. 22, #A38.
D. McKee, Taxicab numbers, Apr 24 2001.
J. C. Meyrignac, The Taxicab Problem
Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
I. Peterson, Math Trek, Taxicab Numbers
Randall L. Rathbun, Sixth Taxicab Number?, posting to the NMBRTHRY mailing list, Jul 16 2002.
W. Schneider, Taxicab Numbers
J. Silverman, Taxicabs and Sums of Two Cubes, American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.
Po-Chi Su, More Upper Bounds on Taxicab and Cabtaxi Numbers, Journal of Integer Sequences, 19 (2016), #16.4.3.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Taxicab Number
Wikipedia, Taxicab number
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.
D. W. Wilson, Taxicab Numbers (last snapshot available on web.archive.org, as of June 2013).
FORMULA
a(n) <= A080642(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013
a(n) > 113*n^3 for n > 1 (a trivial bound based on the number of available cubes; 113 < (1 - 2^(-1/3))^(-3)). - Charles R Greathouse IV, Jun 18 2024
EXAMPLE
From Zak Seidov, Mar 22 2013: (Start)
Values of {b,c}, a(n) = b^3 + c^3:
n = 1: {1,1}
n = 2: {1, 12}, {9, 10}
n = 3: {167, 436}, {228, 423}, {255, 414}
n = 4: {2421, 19083}, {5436, 18948}, {10200, 18072}, {13322, 16630}
n = 5: {38787, 365757}, {107839, 362753}, {205292, 342952}, {221424, 336588}, {231518, 331954}
n = 6: {582162, 28906206}, {3064173, 28894803}, {8519281, 28657487}, {16218068, 27093208}, {17492496, 26590452}, {18289922, 26224366}. (End)
CROSSREFS
KEYWORD
nonn,nice,hard,more
AUTHOR
EXTENSIONS
Added a(6), confirmed by Uwe Hollerbach, communicated by Christian Schroeder, Mar 09 2008
STATUS
approved