# Definition:Hardy-Ramanujan Number

## Contents

## Definition

The $n$th **Hardy-Ramanujan number** $\operatorname {Ta} \left({n}\right)$ is the smallest positive integer which can be expressed as the sum of $2$ cubes in $n$ different ways.

### Sequence of Hardy-Ramanujan numbers

The sequence of Hardy-Ramanujan numbers begins:

- $2, 1729, 87 \, 539 \, 319, 6 \, 963 \, 472 \, 309 \, 248, 48 \, 988 \, 659 \, 276 \, 962 \, 496, \ldots$

## Examples

### $1729$: Hardy-Ramanujan Number $\operatorname{Ta} \left({2}\right)$

The $2$nd Hardy-Ramanujan number $\operatorname {Ta} \left({2}\right)$ is $1729$:

\(\displaystyle 1729\) | \(=\) | \(\displaystyle 12^2 + 1^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 10^2 + 9^2\) |

### $87 \, 539 \, 319$: Hardy-Ramanujan Number $\operatorname{Ta} \left({3}\right)$

The $3$rd Hardy-Ramanujan number $\operatorname {Ta} \left({3}\right)$ is $87 \, 539 \, 319$:

\(\displaystyle 87 \, 539 \, 319\) | \(=\) | \(\displaystyle 167^3 + 436^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 228^3 + 423^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 255^3 + 414^3\) |

### $6 \, 963 \, 472 \, 309 \, 248$: Hardy-Ramanujan Number $\operatorname{Ta} \left({4}\right)$

The $4$th Hardy-Ramanujan number $\operatorname {Ta} \left({4}\right)$ is $6 \, 963 \, 472 \, 309 \, 248$:

\(\displaystyle 6 \, 963 \, 472 \, 309 \, 248\) | \(=\) | \(\displaystyle 2421^3 + 19 \, 083^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 5436^3 + 18 \, 948^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 10 \, 200^3 + 18 \, 072^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 13 \, 322^3 + 16 \, 630^3\) |

## Also known as

The **Hardy-Ramanujan numbers** are also (more commonly) known as **taxicab numbers** from the often-cited anecdote of Hardy's visit to Ramanujan in hospital.

Hence the usual denotation of the $n$th such number: $\operatorname {Ta} \left({n}\right)$; **Ta** for **taxicab**.

However, the name **taxicab number** is ambiguous; it is also defined (with exactly the same nomenclative derivation) as the sequence of numbers expressible as the sum of $2$ positivecubes.

Because of that ambiguity, the term **Hardy-Ramanujan numbers** is to be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in preference.

## Also see

- Definition:Taxicab Number
- Results about
**Hardy-Ramanujan numbers**can be found here.

## Source of Name

This entry was named for Godfrey Harold Hardy and Srinivasa Ramanujan.

## Historical Note

The anecdote related by G.H. Hardy about a visit to Srinivasa Ramanujan in hospital in a taxicab whose number was $1729$ is well-known and often repeated.

The concept was first introduced by Bernard Frénicle de Bessy in $1657$, who discovered $5$ instances of these numbers, including $1729$, in response to a challenge by Leonhard Paul Euler.

Those are the numbers referred to as **taxicab numbers** on $\mathsf{Pr} \infty \mathsf{fWiki}$, following the lead of N.J.A. Sloane on the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

However, the deeper concept of the **Hardy-Ramanujan numbers** is more recent.

After $1729$, the next **Hardy-Ramanujan number** $\operatorname {Ta} \left({3}\right)$ was discovered by John Leech in $1957$ to be $87 \, 539 \, 319$.

## Sources

- Weisstein, Eric W. "Taxicab Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/TaxicabNumber.html