Definition:Hardy-Ramanujan Number
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 $\map {\operatorname {Ta}} 2$ is $1729$:
| \(\ds 1729\) | \(=\) | \(\ds 12^3 + 1^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10^3 + 9^3\) |
$87 \, 539 \, 319$: Hardy-Ramanujan Number $\operatorname{Ta} \left({3}\right)$
The $3$rd Hardy-Ramanujan number $\map {\mathrm {Ta} } 3$ is $87 \, 539 \, 319$:
| \(\ds 87 \, 539 \, 319\) | \(=\) | \(\ds 167^3 + 436^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 228^3 + 423^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 255^3 + 414^3\) |
$6 \, 963 \, 472 \, 309 \, 248$: Hardy-Ramanujan Number $\operatorname{Ta} \left({4}\right)$
The $4$th Hardy-Ramanujan number $\operatorname {Ta} \left({4}\right)$ is $6 \, 963 \, 472 \, 309 \, 248$:
| \(\ds 6 \, 963 \, 472 \, 309 \, 248\) | \(=\) | \(\ds 2421^3 + 19 \, 083^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5436^3 + 18 \, 948^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10 \, 200^3 + 18 \, 072^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 \, 322^3 + 16 \, 630^3\) |
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. https://mathworld.wolfram.com/TaxicabNumber.html