# 1729

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## Number

$1729$ (one thousand, seven hundred and twenty-nine) is:

$7 \times 13 \times 19$

The $1$st taxicab number: a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways:
$1729 = 12^3 + 1^3 = 10^3 + 9^3$

The $1$st Fermat pseudoprime to each of the bases $2$, $3$ and $5$:
$2^{1729} \equiv 2 \pmod {1729}$, $3^{1729} \equiv 3 \pmod {1729}$, $5^{1729} \equiv 5 \pmod {1729}$

The $2$nd Hardy-Ramanujan number after $2$: the smallest positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways:
$1729 = \map {\operatorname {Ta} } 2 = 12^3 + 1^3 = 10^3 + 9^3$

The $3$rd Carmichael number after $561$, $1105$:
$\forall a \in \Z: a \perp 1729: a^{1728} \equiv 1 \pmod {1729}$

The $4$th and largest, after $1$, $81$, $1458$ of the $4$ harshad numbers which can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits:
$1729 = 91 \times 19 = 91 \times \paren {1 + 7 + 2 + 9}$

The $6$th Poulet number after $341$, $561$, $645$, $1105$, $1387$:
$2^{1729} \equiv 2 \pmod {1729}$: $1729 = 7 \times 13 \times 19$

The $7$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$, $561$, $781$, $1541$:
$5^{1729} \equiv 5 \pmod {1729}$

The $9$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$:
$3^{1729} \equiv 3 \pmod {1729}$

The $364$th harshad number:
$1729 = 91 \times 19 = 91 \times \paren {1 + 7 + 2 + 9}$

## Historical Note

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.