17

Number
$17$ (seventeen) is:


 * The $7$th prime number, after $2, 3, 5, 7, 11, 13$


 * The $1$st of the $4$th pair of twin primes, with $19$


 * The $3$rd Fermat number and Fermat prime after $3, 5$:
 * $17 = 2^{2^2} + 1$


 * The $2$nd emirp after $13$.


 * The $3$rd Stern number after $1, 3$.


 * The smallest integer to be the sum of $2$ distinct powers of $4$:
 * $17 = 1 + 16 = 1^4 + 2^4$


 * The only prime number to equal the sum of the digits of its cube:
 * $17 = 4 + 9 + 1 + 3$, while $17^3 = 4913$


 * The smallest odd positive integer not expressible in the form $2 n^2 + p$ where $p$ is prime.


 * The $2$nd prime number whose period is of maximum length:
 * $\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$


 * The $5$th of the lucky numbers of Euler after $2, 3, 5, 11$:
 * $n^2 + n + 17$ is prime for $0 \le n < 15$.


 * The $12$th after $1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

Also see

 * Construction of Regular Heptadecagon
 * Prime Equal to Sum of Digits of Cube
 * Positive Integers Equal to Sum of Digits of Cube
 * Smallest Odd Number not of form 2 a squared plus p
 * Period of Reciprocal of 17 is of Maximal Length