A080642 -id:A080642 - cp's OEIS frontend

A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344

Offset: 1

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

			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)

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.

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
Christian Boyer, New upper bounds on Taxicab and Cabtaxi numbers
Christian Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6.
"Durango" Bill Butler, Durango Bill's Ramanujan Numbers and The Taxicab Problem
Cristian S. Calude, Elena Calude and Michael J. Dinneen, What is the value of Taxicab(6)?
Cristian S. Calude, Elena Calude and Michael J. Dinneen, What is the value of Taxicab(6)?, J. Universal Computer Science, 9 (2003), 1196-1203.
Pavel Emelyanov, On Hunting for Taxicab Numbers, arXiv:0802.1147 [math.NT], 2008.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
U. Hollerbach, The sixth taxicab number is 24153319581254312065344, posting to the NMBRTHRY mailing list, Mar 09 2008.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Dave McKee, Taxicab numbers, Apr 24 2001.
Jean-Charles Meyrignac, The Taxicab Problem
Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
Ivars Peterson, Math Trek, Taxicab Numbers
Randall L. Rathbun, Sixth Taxicab Number?, posting to the NMBRTHRY mailing list, Jul 16 2002.
Walter Schneider, Taxicab Numbers
Joseph H. 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).

Cf. A001235, A003826, A023050, A047696, A080642 (cubefree taxicab numbers).

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

Added a(6), confirmed by Uwe Hollerbach, communicated by Christian Schroeder, Mar 09 2008

A230564 Rational rank of the n-th taxicab elliptic curve x^3 + y^3 = A011541(n).

Original entry on oeis.org

0, 2, 4, 5, 4

Offset: 1

Jonathan Sondow, Oct 25 2013

Guy, 2004: "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively."
Abhinav Kumar computed that a(1) = 0 (see the MathOverflow link for details). But Euler and Legendre scooped him (see the next comment).
Noam D. Elkies: "... the fact that x^3+y^3=2 has no [rational] solutions other than x=y=1 is attributed by Dickson to Euler himself: see Dickson's History of the Theory of Numbers (1920) Vol.II, Chapter XXI "Numbers the Sum of Two Rational Cubes", page 572. The reference (footnote 182) is "Algebra, 2, 170, Art. 247; French transl., 2, 1774, pp. 355-60; Opera Omnia, (1), I, 491". In the next page Dickson also refers to work of Legendre that includes this result (footnote 184: "Théorie des nombres, Paris, 1798, 409; ...")." See the MathOverflow link for further comments from Elkies.

			rank(x^3 + y^3 = 2) = 0.
rank(x^3 + y^3 = 1729) = 2.
rank(x^3 + y^3 = 87539319) = 4.
rank(x^3 + y^3 = 6963472309248) = 5.
rank(x^3 + y^3 = 48988659276962496) = 4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, D1.

MathOverflow, What is the rational rank of the elliptic curve x^3 + y^3 = 2?, Oct 25 2013.
J. Silverman, Taxicabs and sums of two cubes, Amer. Math. Monthly, 100 (1993), 331-340.

Cf. A011541, A060838, A080642.

a(n) = A060838(A011541(n)).

cp's OEIS Frontend

A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

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A272885 Cubefree taxi-cab numbers.

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A230564 Rational rank of the n-th taxicab elliptic curve x^3 + y^3 = A011541(n).

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