taxicab number conference

8/2/2019 Taxicab Number Conference

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Taxicab Number andInteresting Aspects

Presented by Kangkeun Rhee

March 13th, 2010 at Pepperdine University


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Presentation Outline

Two mathematicians to introduce

The story behind Taxicab Numbers

What is a Taxicab Number?

So-far discovered Taxicab Numbers

Aspects about Taxicab Numbers

Cabtaxi Numbers and Information


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Two Mathematicians to introduce

Ramanujan (12/22/1887 to 04/26/1920)

- Highly celebrated mathematician in India, and self-taught

genius with no training in pure mathematics. Mostly seen in

number theory, series, and functions

G. H. Hardy (02/07/1887 to 12/01/1947)

- Number theory without Hardy is like a car without an engine.

-"If I could prove by logic that you would die in five minutes, I

should be sorry you were going to die, but my sorrow wouldbe very much mitigated by pleasure in the proof"


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One Story Behind Taxicab Number

Taxicab with a number of 1729.

- I looked at this number and apparently, I have not seen

anything interesting, says Hardy.

- Actually it is an interesting number BECAUSE it is thesmallest number expressible as the sum of two positive

cubes in two different ways, says Ramanujan.


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So what exactly is a Taxicab Number?

In mathematics, the nth taxicab number, generallydenoted as Ta(n) or Taxicab(n), is the smallest number

that can be expressed as a sum of two positive cubes in

n distinct ways.

So far 6 Taxicab numbers are discovered. Taxicabnumber 7 has not been discovered yet as of now.

The last Taxicab Number was discovered just two years

ago by Hollerbach.

Ramanujan G. H. Hardy


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Discovered Taxicab Numbers

2 discovered by Hardy and Ramanujan, called Hardy-Ramanujan Number

3 discovered by John Leech in 1957, 4 discovered by Rosenstiel in 1993

5 discovered by Dardis 1994, confirmed by David Wilson in 1999

6 announced by Hollerbach, March 8, 2008, following the work done by

Calude in 2003


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Interesting Aspects of Taxicab

Numbers Unfortunately, Hardy and Ramanujan are not the first

publishers of 1729. Bernard Bessy is the first publisher. (1657)

It is quite clear that most of the Taxicab numbers arediscovered in the last century. One thing to note is that Ta(4),

Ta(5), and Ta(6) were discovered in the last few decades.

- Advancement in technology such as calculators andcomputers

Even with technology, Taxicab number is hard to discover.

Although it is proven that the nth Taxicab number exists(Proven by Hardy and Wright), the proof is no use in findingthe number.


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There is also a Cabtaxi Number What is a Cabtaxi Number?

- Cabtaxi number is the smallest positive number that can

be written in n ways as a sum of two (does not have to

be positive cubes, cubes can also be negative) cubes.

- The name is derived from the Taxicab Number(Explained above)

Why cant I be

Either one?


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Cabtaxi Number (1)


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Cabtaxi Number (2)


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Cabtaxi Number (3)


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Information about Cabtaxi No.

Cabtaxi number 5, 6, 7 were found by Randall Rathburn.

Cabtaxi number 8 was discovered by Daniel Bernstein.

Cabtaxi 9, following the method from Daniel Bernstein,

was discovered by Duncan Moore

Christian Boyer, four years ago, reported Cabtaxi 10 to

be an upper bound and Hollerbach later verified to be

Cabtaxi 10 in 2008.


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Personal Acknowledgements

Dr. Brian Fisher of Pepperdine University

Dr. Kendra Killpatrick of Pepperdine University

People who attended the Tuesday Tea Talk on February

16th

Pacific Coast Math Conference

taxicab number conference

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