1729

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:
 * $1729 = 91 \times 19 = 91 \times \paren {1 + 7 + 2 + 9}$

Also see

 * Smallest Fermat Pseudoprime to Bases 2, 3 and 5