Definition:Hardy-Ramanujan Number

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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


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