1729/Historical Note

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Historical Note on $1729$ (One Thousand, Seven Hundred and Twenty-Nine)

In the opinion of some writers, "among the most famous of all numbers".

This is all down to the influence of the writings of G.H. Hardy, who documents an anecdote about a time when he visited Srinivasa Ramanujan in hospital.

He reports the incident thus:

[ Ramanujan ] could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember going to see him once when he was lying ill in Putney. I had ridden in a taxi-cab No. $1729$, and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavourable omen. 'No,' he reflected, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'
$1729 = 12^3 + 1^3 = 10^3 + 9^3$

Because of this, $1729$ is often seen referred to as a taxicab number (sometimes hyphenated: taxi-cab).

Hardy then asked Ramanujan whether he knew the answer to the same problem for $4$th powers. Ramanujan thought for a moment, then said he did not, but he believed the number would be very large.

In fact it is $635 \, 318 \, 657$.

This property of $1729$ appears occasionally in mainstream entertainment either as a subject of mathematics as discussed by supposed mathematicians, or (in more subtle fare) as a mathematical in-joke.