Definition:Taxicab Number
Definition
A taxicab number is a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways.
Sequence of Taxicab Numbers
The sequence of taxicab numbers (sums of $2$ positive cubes) begins:
- $1729, 4104, 13 \, 832, 20 \, 683, 32 \, 832, 39 \, 312, 40 \, 033, 46 \, 683, 64 \, 232, 65 \, 728, 110 \, 656, \ldots$
Also defined as
The usual definition of the $n$th taxicab number is the smallest positive integer which can be expressed as the sum of $2$ cubes in $n$ different ways.
However, the name Hardy-Ramanujan number is also used for that specific concept, which is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ to reduce ambiguity and confusion.
Also known as
The name is sometimes hyphenated: taxi-cab number.
Also see
- Definition:Hardy-Ramanujan Number
- Results about taxicab numbers can be found here.
Historical Note
The first person to find an integer with this property was Bernard Frénicle de Bessy in $1657$.
He discovered $5$ instances of these numbers in response to a challenge by Leonhard Paul Euler:
| \(\ds 1729\) | \(=\) | \(\, \ds 10^3 + 9^3 \, \) | \(\, \ds = \, \) | \(\ds 12^3 + 1^3\) | ||||||||||
| \(\ds 4104\) | \(=\) | \(\, \ds 15^3 + 9^3 \, \) | \(\, \ds = \, \) | \(\ds 16^3 + 2^3\) | ||||||||||
| \(\ds 39 \, 312\) | \(=\) | \(\, \ds 15^3 + 33^3 \, \) | \(\, \ds = \, \) | \(\ds 34^3 + 2^3\) | ||||||||||
| \(\ds 40 \, 033\) | \(=\) | \(\, \ds 16^3 + 33^3 \, \) | \(\, \ds = \, \) | \(\ds 34^3 + 9^3\) | ||||||||||
| \(\ds 20 \, 683\) | \(=\) | \(\, \ds 24^3 + 19^3 \, \) | \(\, \ds = \, \) | \(\ds 27^3 + 10^3\) |
The name taxicab number arises from an anecdote related by G.H. Hardy about a visit to Srinivasa Ramanujan in hospital in a taxicab whose number was $1729$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1729$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1729$