Taxicab Numbers

© Walter Schneider 2000
(last updated 3/2/2003)


 

The name taxicab number or Hardy-Ramanujan number is associated with the following well-known story involving G.H. Hardy and S. Ramanujan:

I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable 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.

Generalizing the above observation the nth taxicab number Ta(n) is defined as the least integer expressible in n different ways as the sum of two positive cubes. In the famous book An Introduction to the Theory of Numbers of Hardy and Wright it is proven that the nth taxicab number exists but the proof is of no use in finding the number.

The table below shows all known taxicab numbers (Sloane's A011541) up to taxicab number Ta(5) found by D.W. Wilson in November 1997. Wilson also found a 6-way sum showing that the 6th taxicab number Ta(6) is ≤ 8230545258248091551205888 = 8.2...·1024. In 1998, D.J. Bernstein showed that Ta(6) ≥ 1018 and in 2002, Randall L. Rathbun discovered the 6-way sum 24153319581254312065344 = 2.4...·1023. Ta(6) is therefore restricted to 1018 ≤ Ta(6) ≤ 2.4...·1023.

When generalizing to higher powers the situation gets even tougher and it's not known if n-way sums for higher powers exist. For fourth powers the smallest 2-way sum is known to be

635318657 = 594 + 1584 = 1334 + 1344.

No example is known for 3-way sums or more. For fifth powers no example of a 2-way sum is known.

Sometimes the definition of taxicab numbers is relaxed by allowing both positive and negative cubes. The smallest n-way sums are called cabtaxi numbers Ca(n) (Sloane's A047696) and the sequence is known up to Ca(8). For Ca(9) it's known that 1019 ≤ Ca(9) ≤ 10933313592720956472 = 1.0...·1019.

Table of known Taxicab Numbers
n Ta(n) #Digits Found by
1 2
= 13 + 13
1 trivial
2 1729
= 13 + 123
= 93 + 103
4 Frenicle de Bessy, 1657
3 87539319
= 1673 + 4363
= 2283 + 4233
= 2553 + 4143
8 Leech, 1957
4 6963472309248
= 24213 + 190833
= 54363 + 189483
= 102003 + 180723
= 133223 + 166303
13 Rosenstiel, Dardis and Rosenstiel, 1991
5 48988659276962496
= 387873 + 3657573
= 1078393 + 3627533
= 2052923 + 3429523
= 2214243 + 3365883
= 2315183 + 3319543
17 David W. Wilson, 1997
Table of known Cabtaxi Numbers
n Ca(n) #Digits Found by
1 0
= 13 - 13
1 trivial
2 91
= 33 + 43
= 63 - 53
2 trivial
3 728
= 63 + 83
= 93 - 13
= 123 - 103
3 trivial
4 2741256
= 1083 + 1143
= 1403 - 143
= 1683 - 1263
= 2073 - 1833
7 ???
5 6017193
= 1663 + 1133
= 1803 + 573
= 1853 - 683
= 2093 - 1463
= 2463 - 2073
7 Randall Rathbun
6 1412774811
= 9633 + 8043
= 11343 - 3573
= 11553 - 5043
= 12463 - 8053
= 21153 - 20043
= 47463 - 47253
10 Randall Rathbun
7 11302198488
= 19263 + 16083
= 19393 + 15893
= 22683 - 7143
= 23103 - 10083
= 24923 - 16103
= 42303 - 40083
= 94923 - 94503
11 Randall Rathbun
8 137513849003496
= 229443 + 500583
= 365473 + 445973
= 369843 + 442983
= 521643 - 164223
= 531303 - 231843
= 573163 - 370303
= 972903 - 921843
= 2183163 - 2173503
15 D.J. Bernstein

 

References

Bernstein, D.J.: Enumerating solutions to p(a)+q(b)=r(c)+s(d),
Mathematics of Computation 70 (233), 2001, pp. 389-394.
euler.free.fr: The Taxicab Problem,
http://euler.free.fr/taxicab.htm.
Hardy, G.H. and Wright, E.M.: An Introduction to the Theory of Numbers,
Oxford University Press, London & NY 1954, 3rd edition, Theorem 412.
Leech, J.: Some Solutions of Diophantine Equations,
Proc. Cambridge Phil. Soc. 53, 1957, pp. 778-780.
Rosenstiel, E. and Dardis, J.A. and Rosenstiel, C.R.: The four least solutions in distinct positive integers of the Diophantine equation s = x3 + y3 = u3 + v3 = m3 + n3 ,
Bulletin Inst. Math. Appl. 27, 1991, pp. 155-157,
http://www.cix.co.uk/~rosenstiel/cubes/welcome.htm.
Silverman, J.H.: Taxicabs and sums of two cubes,
American Mathematical Monthly 100, 1993, pp. 331-340.
Weisstein, Eric W.: Taxicab Number,
CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC 1999.
Wilson, David W.: The Fifth Taxicab Number is 48988659276962496,
Journal of Integer Sequences 2, 1999, Article 99.1.9,
http://www.research.att.com/~njas/sequences/JIS/wilson10.html.
 
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