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%I #148 Jan 30 2023 11:47:17
%S 2,1729,87539319,6963472309248,48988659276962496,
%T 24153319581254312065344
%N Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.
%C 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).
%C A001235 gives another definition of "taxicab numbers".
%C _David W. Wilson_ reports a(6) <= 8230545258248091551205888. [But see next line!]
%C _Randall L Rathbun_ has shown that a(6) <= 24153319581254312065344.
%C 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.
%C 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
%C a(7) <= 24885189317885898975235988544. - _Robert G. Wilson v_, Nov 18 2012
%C 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
%C a(9) <= 136897813798023990395783317207361432493888. - _PoChi Su_, May 17 2013
%C From _PoChi Su_, Oct 09 2014: (Start)
%C The preceding bounds are not the best that are presently known.
%C An upper bound for a(22) was given by C. Boyer (see the C. Boyer link), namely
%C 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.
%C 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
%C 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,
%C 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.
%C (End)
%C From _Sergey Pavlov_, Mar 01 2017: (Start)
%C Let f(n) be a(n). For 1 < n <= 6, f(n) can be written as the product of not more than x(n) distinct prime powers, where x(n) < x(n+1), 2 < x(n) <= 2n, and one of the factors is a power of 7, while, for n > 2, the second factor is 3^3. Additionally, for 1 < n < 6, f(n) can be represented as the difference between two squares (b(n))^2 - (c(n))^2, where b(n) and c(n) are integer, b(n) < b(n+1), c(n) < c(n+1):
%C f(2)=7 *13 *19 = 55^2 - 36^2,
%C f(3)=3^3 *7 *31 *67 *223 = 9788^2 - 2875^2
%C f(4)=2^10 *3^3 *7 *13 *19 *31 *37 *127 = 2638848^2 - 6816^2
%C f(5)=2^6 *3^3 *7^4 *13 *19 *43 *73 *97 *157 = 221334064^2 - 329560^2
%C f(6)=2^6 *3^3 *7^4 *13 *19 *43 *73 *79^3 *97 *157
%C Conjecture: let f(n) be a(n). Then, for n > 1, f(n) can be represented as the product of not more than x(n) distinct prime powers, where x(n) <= x(n+1), 2 < x(n) <= 2n; additionally, while n > 1, f(n) can be written as the difference between two squares (b(n))^2 - (c(n))^2, where b(n) and c(n) are integer, b(n) < b(n+1), c(n) < c(n+1). For n > 3, there are y "old" distinct prime powers o(1)...o(y) such that one of them is a power of 7 and the other is either a power of 3, or 3^3, and z "new" distinct prime powers n(1)...n(z) such that none of them - unlike the "old" ones - can be a divisor of a(q) while q < n.
%C (End)
%D C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
%D R. K. Guy, Unsolved Problems in Number Theory, D1.
%D 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 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.
%H D. J. Bernstein, <a href="http://pobox.com/~djb/papers/sortedsums.dvi">Enumerating solutions to p(a) + q(b) = r(c) + s(d)</a>
%H D. Bill, <a href="http://www.durangobill.com/Ramanujan.html">Durango Bill's Ramanujan Numbers and The Taxicab Problem</a>
%H C. Boyer, <a href="http://www.christianboyer.com/taxicab">New upper bounds on Taxicab and Cabtaxi numbers</a>
%H C. Boyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Boyer/boyer.html">New upper bounds for Taxicab and Cabtaxi numbers</a>, JIS 11 (2008) 08.1.6
%H C. S. & E. Calude and M. T. Dinneen, <a href="http://web.archive.org/web/20040121183032/http://www.jucs.org/jucs_9_10/what_is_the_value/paper.html">What is the value of Taxicab(6)?</a>
%H C. S. Calude, E. Calude and M. J. Dinneen, <a href="https://www.researchgate.net/publication/37987704_What_is_the_Value_of_Taxicab6">What is the value of Taxicab(6)?</a>, J. Universal Computer Science, 9 (2003), 1196-1203.
%H U. Hollerbach, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;f1ac1754.0803">The sixth taxicab number is 24153319581254312065344</a>, posting to the NMBRTHRY mailing list, Mar 09 2008
%H Bernd C. Kellner, <a href="https://arxiv.org/abs/1902.11283">On primary Carmichael numbers</a>, arXiv:1902.11283 [math.NT], 2019. See also <a href="http://math.colgate.edu/~integers/w38/w38.pdf">Integers</a> (2022) Vol. 22, #A38.
%H D. McKee, <a href="http://everything2.net/node/1028223">Taxicab numbers</a>, Apr 24 2001
%H J. C. Meyrignac, <a href="http://euler.free.fr/taxicab.htm">The Taxicab Problem</a>
%H Ken Ono and Sarah Trebat-Leder, <a href="http://arxiv.org/abs/1510.00735">The 1729 K3 surface</a>, arXiv:1510.00735 [math.NT], 2015.
%H I. Peterson, Math Trek, <a href="https://www.sciencenews.org/article/taxicab-numbers">Taxicab Numbers</a>
%H Randall L. Rathbun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;76cec300.0207">Sixth Taxicab Number?</a>, posting to the NMBRTHRY mailing list, Jul 16 2002
%H W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/number-theory/taxicab-numbers.html">Taxicab Numbers</a>
%H J. Silverman, <a href="http://www.jstor.org/stable/2324954">Taxicabs and Sums of Two Cubes</a>, American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.
%H Po-Chi Su, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Su/su3.html">More Upper Bounds on Taxicab and Cabtaxi Numbers</a>, Journal of Integer Sequences, 19 (2016), #16.4.3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TaxicabNumber.html">Taxicab Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Taxicab_number">Taxicab number</a>
%H D. W. Wilson, <a href="https://cs.uwaterloo.ca/journals/JIS/wilson10.html">The Fifth Taxicab Number is 48988659276962496</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.9.
%H D. W. Wilson, <a href="http://web.archive.org/web/20130602112401/http://pi.lacim.uqam.ca/eng/problem_en.html">Taxicab Numbers</a> (last snapshot available on web.archive.org, as of June 2013).
%F a(n) <= A080642(n) for n > 0, with equality for n = 1, 2 (only?). - _Jonathan Sondow_, Oct 25 2013
%e From _Zak Seidov_, Mar 22 2013: (Start)
%e Values of {b,c}, a(n) = b^3 + c^3:
%e n = 1: {1,1}
%e n = 2: {1, 12}, {9, 10}
%e n = 3: {167, 436}, {228, 423}, {255, 414}
%e n = 4: {2421, 19083}, {5436, 18948}, {10200, 18072}, {13322, 16630}
%e n = 5: {38787, 365757}, {107839, 362753}, {205292, 342952}, {221424, 336588}, {231518, 331954}
%e n = 6: {582162, 28906206}, {3064173, 28894803}, {8519281, 28657487}, {16218068, 27093208}, {17492496, 26590452}, {18289922, 26224366}. (End)
%Y Cf. A001235, A003826, A023050, A047696, A080642 (cubefree taxicab numbers).
%K nonn,nice,hard,more
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E Added a(6), confirmed by Uwe Hollerbach, communicated by _Christian Schroeder_, Mar 09 2008
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