Taxicab number

In mathematics , the -th taxicab number is defined as the smallest (natural) number that can be represented in various ways as the sum of two cubic numbers . Godfrey Harold Hardy and EM Wright have shown that for every natural number there is a taxicab number. However, the evidence says nothing about the occurrence of these numbers, so they can only be found with great (computer-aided) effort. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

It owes its name to a famous anecdote by Hardy. He visited Ramanujan at the bedside and mentioned that he had come in a taxi number 1729, which Hardy thought was an uninteresting number . Ramanujan did not find this by showing Hardy the properties mentioned above.

Well-known taxicab numbers

The following six taxicab numbers are known (sequence A011541 in OEIS ):

${\displaystyle {\begin{aligned}\operatorname {Ta} (1)=2&=1^{3}+1^{3}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Ta} (2)=1729&=1^{3}+12^{3}\\&=9^{3}+10^{3}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Ta} (3)=87539319&=167^{3}+436^{3}\\&=228^{3}+423^{3}\\&=255^{3}+414^{3}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Ta} (4)=6963472309248&=2421^{3}+19083^{3}\\&=5436^{3}+18948^{3}\\&=10200^{3}+18072^{3}\\&=13322^{3}+16630^{3}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Ta} (5)=48988659276962496&=38787^{3}+365757^{3}\\&=107839^{3}+362753^{3}\\&=205292^{3}+342952^{3}\\&=221424^{3}+336588^{3}\\&=231518^{3}+331954^{3}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Ta} (6)=24153319581254312065344&=582162^{3}[phone]^{3}\\&=3064173^{3}[phone]^{3}\\&=8519281^{3}[phone]^{3}\\&=16218068^{3}[phone]^{3}\\&=17492496^{3}[phone]^{3}\\&=18289922^{3}[phone]^{3}\end{aligned}}}$

Upper bounds on taxicab numbers

Upper bounds are known for the following six taxicab numbers:

${\displaystyle {\begin{matrix}\operatorname {Ta} (7)&\leq &24885189317885898975235988544&=&2648660966^{3}[phone]^{3}\\&&&=&2685635652^{3}[phone]^{3}\\&&&=&2736414008^{3}[phone]^{3}\\&&&=&2894406187^{3}[phone]^{3}\\&&&=&2915734948^{3}[phone]^{3}\\&&&=&2918375103^{3}[phone]^{3}\\&&&=&2919526806^{3}[phone]^{3}\end{matrix}}}$

${\displaystyle {\begin{matrix}\operatorname {Ta} (8)&\leq &50974398750539071400590819921724352&=&299512063576^{3}[phone]^{3}\\&&&=&336379942682^{3}[phone]^{3}\\&&&=&341075727804^{3}[phone]^{3}\\&&&=&347524579016^{3}[phone]^{3}\\&&&=&367589585749^{3}[phone]^{3}\\&&&=&370298338396^{3}[phone]^{3}\\&&&=&370633638081^{3}[phone]^{3}\\&&&=&370779904362^{3}[phone]^{3}\end{matrix}}}$

${\displaystyle {\begin{matrix}\operatorname {Ta} (9)&\leq &136897813798023990395783317207361432493888&=&41632176837064^{3}[phone]^{3}\\&&&=&46756812032798^{3}[phone]^{3}\\&&&=&47409526164756^{3}[phone]^{3}\\&&&=&48305916483224^{3}[phone]^{3}\\&&&=&51094952419111^{3}[phone]^{3}\\&&&=&51471469037044^{3}[phone]^{3}\\&&&=&51518075693259^{3}[phone]^{3}\\&&&=&51530042142656^{3}[phone]^{3}\\&&&=&51538406706318^{3}[phone]^{3}\end{matrix}}}$

${\displaystyle {\begin{matrix}\operatorname {Ta} (10)&\leq &7335345315241855602572782233444632535674275447104&=&15695330667573128^{3}+15137846555691028^{3}\\&&&=&17627318136364846^{3}+12293996879974082^{3}\\&&&=&17873391364113012^{3}+11757988429199376^{3}\\&&&=&18211330514175448^{3}+10901351979041108^{3}\\&&&=&19262797062004847^{3}+5726433061530961^{3}\\&&&=&19404743826965588^{3}+3058262831974168^{3}\\&&&=&19422314536358643^{3}+2059655218961613^{3}\\&&&=&19426825887781312^{3}+1536982932706676^{3}\\&&&=&19429379778270560^{3}[phone]^{3}\\&&&=&19429979328281886^{3}[phone]^{3}\end{matrix}}}$

${\displaystyle {\begin{matrix}\operatorname {Ta} (11)&\leq &2818537360434849382734382145310807703728251895897826621632&=&11410505395325664056^{3}+11005214445987377356^{3}\\&&&=&12815060285137243042^{3}+8937735731741157614^{3}\\&&&=&12993955521710159724^{3}+8548057588027946352^{3}\\&&&=&13239637283805550696^{3}+7925282888762885516^{3}\\&&&=&13600192974314732786^{3}+6716379921779399326^{3}\\&&&=&14004053464077523769^{3}+4163116835733008647^{3}\\&&&=&14107248762203982476^{3}+2223357078845220136^{3}\\&&&=&14120022667932733461^{3}+1497369344185092651^{3}\\&&&=&14123302420417013824^{3}+1117386592077753452^{3}\\&&&=&14125159098802697120^{3}+657258405504578668^{3}\\&&&=&14125594971660931122^{3}+284485090153030494^{3}\end{matrix}}}$

${\displaystyle {\begin{matrix}\operatorname {Ta} (12)&\leq &73914858746493893996583617733225161086864012865017882136931801625152&=&33900611529512547910376^{3}+32696492119028498124676^{3}\\&&&=&38073544107142749077782^{3}+26554012859002979271194^{3}\\&&&=&38605041855000884540004^{3}+25396279094031028611792^{3}\\&&&=&39334962370186291117816^{3}+23546015462514532868036^{3}\\&&&=&40406173326689071107206^{3}+19954364747606595397546^{3}\\&&&=&41606042841774323117699^{3}+12368620118962768690237^{3}\\&&&=&41912636072508031936196^{3}+6605593881249149024056^{3}\\&&&=&41950587346428151112631^{3}+4448684321573910266121^{3}\\&&&=&41960331491058948071104^{3}+3319755565063005505892^{3}\\&&&=&41965847682542813143520^{3}+1952714722754103222628^{3}\\&&&=&41965889731136229476526^{3}+1933097542618122241026^{3}\\&&&=&41967142660804626363462^{3}+845205202844653597674^{3}\end{matrix}}}$

Discovery story

Ta (2) = 1,729 is also known as the Hardy-Ramanujan number due to the anecdote above ; it was published in 1657 by Bernard Frénicle de Bessy .

Ta (3) = 87,539,319 was discovered by John Leech in 1957 .

Ta (4) was found in 1991 by the amateur number theorist E. Rosenstiel

Ta (5) has been owed to David W. Wilson since 1999. Independently of that, she found Daniel Bernstein a few months later .

Ta (6) was discovered in 2003. Daniel Bernstein had previously specified an upper bound in 1998.

Generalized taxicab number

As a generalized Taxicab numbers is called a modification of the ordinary Taxicab numbers. The definition is:

${\displaystyle \operatorname {Taxicab} (k,j,n)}$is the smallest natural number that can be expressed in various ways using terms of -th powers . ${\displaystyle n}$ ${\displaystyle j}$ ${\displaystyle k}$

For and these are the "normal" taxicab numbers. ${\displaystyle k=3}$ ${\displaystyle j=2}$

Leonhard Euler showed that:

${\displaystyle \mathrm {Taxicab} (4,2,2)=635.318.657=59^{4}+158^{4}=133^{4}+134^{4}}$.

An unsolved problem in mathematics is an existence theorem for values ​​other than and . No solutions were found for these values ​​even with computer assistance. This problem is related to Euler's Conjecture , a generalization of Fermat's Great Theorem . ${\displaystyle k=3}$ ${\displaystyle j=2}$

literature

• Joseph Silverman : Taxicabs and Sums of Two Cubes . In: American Mathematical Monthly Vol. 100, 1993, ISSN  0002-9890 , pp. 331-340.

Web links

• Eric W. Weisstein : Taxicab Number . In: MathWorld (English).
• Taxicab number, from Meyrignac in the Euler network
• Taxicab numbers in the Euler network
• Ivars Peterson on taxicab numbers ( Memento from August 3, 2002 in the Internet Archive )

Individual evidence

1. ^ Godfrey Harold Hardy, Edward Maitland Wright: An introduction to the theory of numbers. Oxford UP, 4th edition 1975, p. 333, Theorem 412, with annotations p. 338f. The first edition is from 1938.
2. ↑ Hardy: Ramanujan , London 1940. Hardy wrote literally:

   “I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways. '”

   Quotations by GH Hardy ( Memento of the original from July 16, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
3. ^ Christian Boyer: New Upper Bounds for Taxicab and Cabtaxi Numbers
4. ↑ Bruce Berndt , S. Bhargava: Ramanujan - For Lowbrows . In: American Mathematical Monthly , Volume 100, 1993, pp. 645-656.
5. ^ J. Leech: Some Solutions of Diophantine Equations. In: Proc. Cambridge Phil. Soc. , 531957, pp. 778-780.
6. ↑ E. Rosenstiel, JA Dardis, CR Rosenstiel: The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation In: Bull. Inst. Math. Appl. , 271991, pp. 155-157. ${\displaystyle s=x^{3}+y^{3}=z^{3}+w^{3}=u^{3}+v^{3}=m^{3}+n^{3}}$
7. ^ DW Wilson: The Fifth Taxicab Number is 48988659276962496 . In: J. Integer Sequences 2, # 99.1.9, 1999.
8. ↑ CS Calude, E. Calude, MJ Dinneen: What Is the Value of Taxicab (6)? (PDF) In: J. Uni. Comp. Sci. , 9, 2003, pp. 1196-1203
9. ^ Richard K. Guy : Unsolved problems in number theory (third edition) . Springer Science and Business Media, New York 2004, ISBN 0-387-20860-7 , p. 437.