A Short Note on TaxiCab and CabTaxi Numbers

Published by Ganit Charcha | Category - Math Articles | 2015-07-20 06:16:51

Pierre de Fermat posed one problem more than 350 years ago and the problem is stated as follows “Find a number which can be written in two different ways as sum of two cube numbers.” The problem can be stated algebraically as follows. To find out integers $x$, $y$, $z$, $w$ such that $$x^{3} + y^{3} = z^{3} + w^{3}$$ The French mathematician Frenicle de Bessy figured out several solutions to the problem and his first solutions were $$1729 = 9^{3} + 10 ^{3} = 1^{3} + 12^{3}$$ and $$4104 = 9^{3} + 15 ^{3} = 2^{3} + 16^{3}$$ And in fact $1729$ it is the smallest number which can be expressed as sum of 2 distinct cube numbers in 2 different ways. The number $1729$ is known to modern mathematician as the Hardy-Ramanujan number. The naming is based on a famous anecdote about Hardy visiting Ramanujan in the hospital. Once, Hardy went to the hospital in Putney to see Ramanujan when he was ill. Hardy has come to the hospital in a taxi whose number was 1729. He told Ramanujan that it was a very dull number. Then Ramanujan pointed out that it is the smallest number which can be expressed as sum of two cubes in two different ways.
In fact, Ramanujan in his works found out a very nice condition of finding integral solutions to the above problem. But, this condition only gives a small fraction of the solutions to the problem.
If $m^{2} + mn + n^{2} = 3a^{2}b$, then $$(m + ab^{2})^{3} + (bn + a)^{3} = (bm + a)^{3} + (n + ab^{2})^{3}$$ With $m = 3$ , $n = 0$, $a = 1$ and $b = 3$, the equation gives $9^{3} + 10 ^{3} = 1^{3} + 12^{3}.$ Ramanujan’s condition is easy to prove. Note that, $(m + ab^{2})^{3} + (bn + a)^{3} - (bm + a)^{3} + (n + ab^{2})^{3}$ $$= (m^{3} - n^{3}) - b^{3}(m^{3} - n^{3}) + 3ab^{2}(m^{2} - n^{2}) + 3 a^{2}b^{4}(m - n) - 3ab^{2}(m^{2} - n^{2}) - 3ba^{2}(m - n)$$$$=(1-b^{3})(m^{3} - n^{3}) - 3ab^{2}(m - n)(1-b^{3})$$$$=(1-b^{3})(m^{3} - n^{3}) - (1-b^{3})(m-n)( m^{2} + mn + n^{2})$$$$ = (1-b^{3})(m^{3} - n^{3}) - (1-b^{3})(m^{3} - n^{3}) = 0$$ Werebrusow also published a similar sort of condition which also helps in finding out integral solutions (invloving both positive and negative integers) to the problem posed by Fermat and the condition is as follows. If $m^{2} + mn + n^{2} = 3a^{2}bc$, then $$((m+n)c + ab^{2})^{3} + (-(m+n)b - ac^{2})^{3} = (-mc + ab^{2})^{3} + (mb - ac^{2})^{3} = (-nc + ab^{2})^{3} + (nb - ac^{2})^{3}$$
$1729$ is also known as one of the TaxiCab numbers. The naming is again based on the same anecdote. $Taxicab(n)$ is defined as the smallest number which can be expressed as sum of $2$ distinct positive integers in $n$ different ways. $1729 = 9^{3} + 10 ^{3} = 1^{3} + 12^{3}$ is $TaxiCab(2)$ and $TaxiCab(1)$ is $2$ ($1^{3} +1^{3} = 2$). $$Taxicab(3) = 87539319 = 167^{3} + 436^{3} = 228^{3} + 423^{3} = 255^{3} + 414^{3}$$Over the 20th century 4 other TaxiCab numbers were found using supercomputer since the numbers were horrendously big. $Taxicab(3)$ was found by John Leech in 1957, $Taxicab(4)$ was found by E.Rosenstiel , J.A. Dardis and C.R. Rosenstiel in 1989 , $Taxicab(5)$ was discovered by J. A. Dardis in 1994 and $Taxicab(6)$ was discovered by R. Rathbun in 2002.
Mathematician didn’t limit the sums to positive cubes. They generalized problem which allows sum of positive or negative cubes. Therefore, $CabTaxi(n)$ is defined as the smallest number which can be expressed as sum of $2$ distinct integers (positive or negative) in $n$ different ways. For example, $CabTaxi(1) = 1 = 1^{3} + 0^{3},$ $$CabTaxi(2) = 91 = 3^{3} + 4^{3} = 6^{3} + (-5)^{3}$$ and $$CabTaxi(3) = 728 = 8^{3} + 6^{3} = 9^{3} + (-1)^{3} = 12^{3} + (-10)^{3}$$ Note that, Werebrusow's condition for $m = 0$ , $n = 3$, $a = 1$, $b = 3$ and $c = 1$ helps in generating $CabTaxi(3)$ as $$12^{3} + (-10)^{3} = 9^{3} + (-1)^{3} = 8^{3} + 6^{3}$$ Value till $CabTaxi(10)$ has been found so far. The interested readers can follow the link http://www.christianboyer.com/taxicab/ to know details about different $TaxiCab$ and $CabTaxi$ numbers.

About the Author: Aratrik Dasgupta is studying in 10th standard in Apeejay School, Noida.  His hobbies include playing football and drawing. Madhubani painting is his favorite pass time. He is passionate about mathematics and would love to explore the world of mathematics, computing and astrophysics. He enjoys solving problem and finding solution in as short time as possible.


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