# 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 cubes in n distinct ways, up to order of summands. G. H. Hardy and E. M. Wright proved in 1954 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 is thus useless in finding Ta(n).

## Known taxicab numbers

So far, the following six taxicab numbers are known (sequence A011541 in OEIS):
$\operatorname{Ta}(1) = 2 = 1^3 + 1^3$
$\begin{matrix}\operatorname{Ta}(2)&=&1729&=&1^3 &+& 12^3 \\&&&=&9^3 &+& 10^3\end{matrix}$
$\begin{matrix}\operatorname{Ta}(3)&=&87539319&=&167^3 &+& 436^3 \\&&&=&228^3 &+& 423^3 \\&&&=&255^3 &+& 414^3\end{matrix}$
$\begin{matrix}\operatorname{Ta}(4)&=&6963472309248&=&2421^3 &+& 19083^3 \\&&&=&5436^3 &+& 18948^3 \\&&&=&10200^3 &+& 18072^3 \\&&&=&13322^3 &+& 16630^3\end{matrix}$
$\begin{matrix}\operatorname{Ta}(5)&=&48988659276962496&=&38787^3 &+& 365757^3 \\&&&=&107839^3 &+& 362753^3 \\&&&=&205292^3 &+& 342952^3 \\&&&=&221424^3 &+& 336588^3 \\&&&=&231518^3 &+& 331954^3\end{matrix}$
$\begin{matrix}\operatorname{Ta}(6)&=&24153319581254312065344&=&582162^3 &+& 28906206^3 \\&&&=&3064173^3 &+& 28894803^3 \\&&&=&8519281^3 &+& 28657487^3 \\&&&=&16218068^3 &+& 27093208^3 \\&&&=&17492496^3 &+& 26590452^3 \\&&&=&18289922^3 &+& 26224366^3\end{matrix}$

## Discovery history

The eccentric British mathematician G.H. Hardy is known for his achievements in number theory and mathematical analysis. But he is perhaps even better known for his adoption and mentoring of the self-taught Indian mathematical genius, Srinivasa Ramanujan.
Hardy himself was a prodigy from a young age, and stories are told about how he would write numbers up to millions at just two years of age, and how he would amuse himself in church by factorizing the hymn numbers. He graduated with honours from Cambridge University, where he was to spend most of the rest of his academic career.
Hardy is sometimes credited with reforming British mathematics in the early 20th Century by bringing a Continental rigour to it, more characteristic of the French, Swiss and German mathematics he so much admired, rather than British mathematics. He introduced into Britain a new tradition of pure mathematics (as opposed to the traditional British forte of applied mathematics in the shadow of Newton), and he proudly declared that nothing he had ever done had any commercial or military usefulness.
Just before the First World War, Hardy (who was given to flamboyant gestures) made mathematical headlines when he claimed to have proved the Riemann Hypothesis. In fact, he was able to prove that there were infinitely many zeroes on the critical line, but was not able to prove that there did not exist other zeroes that were NOT on the line (or even infinitely many off the line, given the nature of infinity).
Meanwhile, in 1913, Srinivasa Ramanujan, a 23-year old shipping clerk from Madras, India, wrote to Hardy (and other academics at Cambridge), claiming, among other things, to have devised a formula that calculated the number of primes up to a hundred million with generally no error. The self-taught and obsessive Ramanujan had managed to prove all of Riemann’s results and more with almost no knowledge of developments in the Western world and no formal tuition. He claimed that most of his ideas came to him in dreams.
Hardy was only one to recognize Ramanujan's genius, and brought him to Cambridge University, and was his friend and mentor for many years. The two collaborated on many mathematical problems, although the Riemann Hypothesis continued to defy even their joint efforts.
 Hardy-Ramanujan "taxicab numbers"
A common anecdote about Ramanujan during this time relates how Hardy arrived at Ramanujan's house in a cab numbered 1729, a number he claimed to be totally uninteresting. Ramanujan is said to have stated on the spot that, on the contrary, it was actually a very interesting number mathematically, being the smallest number representable in two different ways as a sum of two cubes. Such numbers are now sometimes referred to as "taxicab numbers".
It is estimated that Ramanujan conjectured or proved over 3,000 theorems, identities and equations, including properties of highly composite numbers, the partition function and its asymptotics and mock theta functions. He also carried out major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Among his other achievements, Ramanujan identified several efficient and rapidly converging infinite series for the calculation of the value of π, some of which could compute 8 additional decimal places of π with each term in the series. These series (and variations on them) have become the basis for the fastest algorithms used by modern computers to compute π to ever increasing levels of accuracy (currently to about 5 trillion decimal places).
Eventually, though, the frustrated Ramanujan spiralled into depression and illness, even attempting suicide at one time. After a period in a sanatorium and a brief return to his family in India, he died in 1920 at the tragically young age of 32. Some of his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired vast amounts of further research and have have found applications in fields as diverse as crystallography and string theory.
Hardy lived on for some 27 years after Ramanujan’s death, to the ripe old age of 70. When asked in an interview what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, and even called their collaboration "the one romantic incident in my life". However, Hardy too became depressed later in life and attempted suicide by an overdose at one point. Some have blamed the Riemann Hypothesis for Ramanujan and Hardy's instabilities, giving it something of the reputation of a curse.

Ta(2), also known as the Hardy-Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657 and later immortalized by an incident involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy [1]:
 “ 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 [positive] cubes in two different ways." ”
The subsequent taxicab numbers were found with the help of computers. 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.[1][2] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,[3] following a 2003 paper by Calude et al. that gave a 99% chance that the number was actually Ta(6).[4] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.[5]
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 for all pairs (x, y). 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)

## Notes

1. ^ Numbers Count column of Personal Computer World, page 610, Feb 1995
2. ^ "The Fifth Taxicab Number is 48988659276962496" by David W. Wilson
3. ^ NMBRTHRY Archives - March 2008 (#10) "The sixth taxicab number is 24153319581254312065344" by Uwe Hollerbach
4. ^ 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
5. ^ "'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 equation s = x3 + y3 = z3 + w3 = u3 + v3 = m3 + n3, Bull. Inst. Math. Appl., 27(1991) 155-157; MR 92i:11134, online. See also Numbers Count 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's 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