World Library  
Flag as Inappropriate
Email this Article

Taxicab number

Article Id: WHEBN0000513117
Reproduction Date:

Title: Taxicab number  
Author: World Heritage Encyclopedia
Language: English
Subject: Cabtaxi number, Srinivasa Ramanujan, 1729 (number), Generalized taxicab number, Euler's sum of powers conjecture
Collection: Number Theory, Srinivasa Ramanujan
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Taxicab number

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways.

The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:

Contents

  • Definition 1
  • Known taxicab numbers 2
  • Discovery history 3
  • Cubefree taxicab numbers 4
  • See also 5
  • Notes 6
  • References 7
  • External links 8

Definition

The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and thus it cannot be used to find the actual value of Ta(n).

The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.

Known taxicab numbers

So far, the following six taxicab numbers are known (sequence A011541 in OEIS):

\begin{matrix}\operatorname{Ta}(1)&=&2 &=& 1^3 + 1^3\end{matrix}
\begin{matrix}\operatorname{Ta}(2)&=&1729&=&1^3 &+& 12^3 \\&&&=&9^3 &+& 10^3\end{matrix}
\begin{matrix}\operatorname{Ta}(3)&=&87539319&=&167^3 &+& 436^3 \\&&&=&228^3 &+& 423^3 \\&&&=&255^3 &+& 414^3\end{matrix}
\begin{matrix}\operatorname{Ta}(4)&=&6963472309248&=&2421^3 &+& 19083^3 \\&&&=&5436^3 &+& 18948^3 \\&&&=&10200^3 &+& 18072^3 \\&&&=&13322^3 &+& 16630^3\end{matrix}
\begin{matrix}\operatorname{Ta}(5)&=&48988659276962496&=&38787^3 &+& 365757^3 \\&&&=&107839^3 &+& 362753^3 \\&&&=&205292^3 &+& 342952^3 \\&&&=&221424^3 &+& 336588^3 \\&&&=&231518^3 &+& 331954^3\end{matrix}
\begin{matrix}\operatorname{Ta}(6)&=&24153319581254312065344&=&582162^3 &+& 28906206^3 \\&&&=&3064173^3 &+& 28894803^3 \\&&&=&8519281^3 &+& 28657487^3 \\&&&=&16218068^3 &+& 27093208^3 \\&&&=&17492496^3 &+& 26590452^3 \\&&&=&18289922^3 &+& 26224366^3\end{matrix}

Discovery history

Ta(2), also known as the Hardy–Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657.

The subsequent taxicab numbers were found with the help of supercomputers. John Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1991. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.[3][4] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,[5] following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6).[6] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.[7]

Cubefree taxicab numbers

A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 13. When a cubefree taxicab number T is written as T = x3 + y3, the numbers x and y must be relatively prime. Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by Paul Vojta (unpublished) in 1981 while he was a graduate student. It is

15170835645
= 5173 + 24683
= 7093 + 24563
= 17333 + 21523.

The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is

1801049058342701083
= 922273 + 12165003
= 1366353 + 12161023
= 3419953 + 12076023
= 6002593 + 11658843

(sequence A080642 in OEIS).

See also

Notes

  1. ^ Quotations by G. H. Hardy, MacTutor History of Mathematics
  2. ^  
  3. ^ Numbers Count column of Personal Computer World, page 610, Feb 1995
  4. ^ "The Fifth Taxicab Number is 48988659276962496" by David W. Wilson
  5. ^ NMBRTHRY Archives – March 2008 (#10) "The sixth taxicab number is 24153319581254312065344" by Uwe Hollerbach
  6. ^ C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), pp. 1196–1203
  7. ^ "'New Upper Bounds for Taxicab and Cabtaxi Numbers" Christian Boyer, France, 2006–2008

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412.
  • J. Leech, Some Solutions of Diophantine Equations, Proc. Cambridge Phil. Soc. 53, 778–780, 1957.
  • E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equations = x3 + y3 = z3 + w3 = u3 + v3 = m3 + n3, Bull. Inst. Math. Appl., 27(1991) 155–157; MR 92i:11134, online.
  • Numbers Count column, Personal Computer World, November 1989.
  • David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online. (Wilson was unaware of J. A. Dardis' prior discovery of Ta(5) in 1994 when he wrote this.)
  • D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d), Mathematics of Computation 70, 233 (2000), 389–394.
  • C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), p. 1196–1203

External links

  • A 2002 post to the Number Theory mailing list by Randall L. Rathbun
  • Grime, James; Bowley, Roger.  
  • Taxicab and other maths at Euler
  •  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.