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

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• nice
• hard
• more

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

Examples

			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)
		

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)
• 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).

Crossrefs

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

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

Extensions

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