Ramanujan and Taxicab number

Srinivasa Ramanujan, one of the strangest and most miraculous Indian mathematicians and scientists of all time was born 22^nd December in the year 1887.  He had no formal education in mathematics but he made significant contributions to the Mathematics. In his short life span, he invented almost 3,900 results, including identities, theorems and equations. His valuable contributions to mathematics are mainly in number theory, divergent series, mathematical analysis and continued fractions. With the help of V. Ramaswamy Aiyer, Ramanujan first published his work in the “Journal of the Indian Mathematical Society”. In the year 1913, Srinivasa Ramanujan wrote the letter to G. H. Hardy about his works which includes nine pages of manuscript.  On seeing the theorems sent by Ramanujan, G. H. Hardy said that the theorems had “defeated me completely”. In 1914, Ramanujan traveled to England and began collaborative work with G. H. Hardy and Littlewood have been at Cambridge for almost five years. On seeing the Ramanujan notebooks, Hardy said that Ramanujan can compare only with Euler or Jacobi.

Hardy – Ramanujan number

The Hardy – Ramanujan number was invented by two prominent mathematicians, one is the British Mathematician G. H. Hardy and the other one is the Indian genius Srinivasa Ramanujan. The number 1729 is called Hardy – Ramanujan number. The special feature of this number is that “1729 is the smallest number which can be represented in two different ways as the sum of the cubes of two numbers”. This remarkable feature emerged from an incident that occurred during Hardy’s hospital visit to meet Ramanujan, who was ill in 1919. During that visit, Hardy traveled to the hospital by taxi numbered 1729. When he met Ramanujan, he said that the number of the taxi he had ridden was a dull one. But Ramanujan replied, “The number has an interesting feature” and explained it.

Some of the numbers that can be represented in atleast two different ways as the sum of the cubes of two numbers are 4104, 13832, 20683 etc.,

After the death of Ramanujan in the year 1919, it has been proved that these kinds of numbers exist for all positive numbers n by G. H. Hardy and E. M. Wright in 1938. Proof of results can be obtained by using the programming language but the result obtained does not tend to be the smallest number.

Taxicab Number

The smallest positive integer that can be represented as sum of cubes of two positive integers in n different ways is known as n^th taxicab numbers and is denoted by Ta(n). The n^th taxicab number is also known as the n^th Hardy – Ramanujan number.

The generalization of Hardy – Ramanujan number leads to the creation of the idea of taxicab numbers. So far only six taxicab numbers are known.

• The first taxicab number is

• The Hardy – Ramanujan number is the 2^nd taxicab number which is the most famous one. It can be expressed in two different ways as follows.

• In 1957, with the aid of computers, John Leech obtained the 3^rd taxicab number and it can be expressed in three different ways as follows.

• E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel obtained the 4^th taxicab number in the year 1989 and it can be expressed in four different ways as follows.

• In 1994, J. A. Dardis obtained the 5^th taxicab number but it was confirmed in the year 1999 by David W. Wilson and it can be expressed in five different ways as follows.

• In 2008, Uwe Hollerbach announces the 6^th taxicab number and it can be expressed in six different ways as follows.

Cubefree taxicab numbers

A cubefree taxicab number is a number that is not divisible by any other cube except 1³. It is generally denoted by x³+y³, where the numbers x and y should be relatively prime. Among the six taxicab numbers mentioned above, Ta(1) and Ta(2) are the only cubefree taxicab numbers.

• In 1981, Paul Vojta discovered the smallest cubefree taxicab number 15170835645 which can expressed in three different ways.
• In 2003, Stuart Gascoigne and Duncan Moore (independently) discovered the smallest cubefree taxicab number 1801049058342701083 which can expressed in four different ways.