# Taxi Cab number

Cabin number of taxi

Ramanujan's Taxicab Number Driving in a taxi with the number 1729, I noticed that the number seemed rather boring to me and that I was hoping it wasn't a bad name. No, " he answered, "it is a very interesting number; it is the smallest number that can be expressed as the summation of two dice in two different ways.

Ramanujan stressed that 1729 is the smallest number that meets such requirements. In formal terms, and in honour of the Ramanujan-Hardy talk, the smallest number that can be expressed as the total of two dice in different ways is called the taxi cab number and .

However, a generalised taxi cab number can be the smallest number that can be given as the total of a number of entitlements in different ways and as . As an example, since , and is the smallest such number that corresponds to the parameter of , ,, and... Curiously, no one knows what the general taxi cab number, equal for all.

With other words, if one takes away the requirement that the number must be the smallest and we leave, the issue can be reformulated as follows: Is there a number that can be expressed as the summation of two fifth power positives in two different ways?

This is the name given by a mathematician to a number of numbers: 2, 1729 etc. It is the smallest number that can be summed up as the number of two dice in n different ways. This has nothing to do with taxi cars, but the name comes from a well-known discussion between two renowned mathematicians:

Divfrey Hardy and Srinivasa Ramanujan. Hardy Godfrey was appointed to the post of Professors of Mathematics at Cambridge University. He went one of these days to see a young brillant pal, the young India-based mathematician Srinivasa Ramanujan, who was ill. They were both men, and both were men of science and thought about numbers. Ramanujan overheard Hardy come in a taxi and asked him what the taxi number was.

And Hardy said it was just a dull number: 1729. And Ramanujan answered that 1729 was not a dull number at all: it was a very interesting number. It was the smallest number that could be represented by the total of two dice in two different ways. It is a history well known among maths scholars.

The 1729 is sometimes referred to as the "Hardy Ramanujan number". If a number is multiplicated by itself, the response is known as a "square", e.g. 3x3=9, so that the number 9 is a square. 9 is a squared. If a number is multiplicated three and a half by itself, the response is a " dice ", e.g. 3x3x3x3=27, so the number 27 is a dice.

A further example for a dice is 8, because it is 2 x 2 x 2. 27+8=35, i.e. 35 is the "sum of two dice" ("sum" in this context means "numbers that are added"). It is possible to say in two ways that 1729 is the total of two dice. There are other numbers that can be represented as the total of two dice in more than one way, but 1729 is the smallest of them.

And Ramanujan didn't really detect this fact. Ever since the famed talk between Hardy and Ramanujan, philosophers have tried to find other interesting numbers that are the smallest number that can be represented by the total of two dice in three, four, five, etc. in different ways. The following six taxi cab numbers are known so far (sequence 1011541 in the OEIS):