11

Number
$11$ (eleven) is:


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


 * The $1$st of the $3$rd pair of twin primes, with $13$


 * The upper end of the $2$nd record-breaking gap between twin primes:
 * $11 - 7 = 4$


 * The $5$th palindromic prime (after the trivial $1$-digit $2$, $3$, $5$, $7$).


 * The $4$th Sophie Germain prime after $2$, $3$, $5$:
 * $2 \times 11 + 1 = 23$, which is prime.


 * The $2$nd repunit after the trivial case $1$


 * The $1$st repunit prime


 * The only palindromic prime with an even number of digits


 * The $5$th permutable prime after $2$, $3$, $5$, $7$.


 * The smallest number which is the sum of a square and a prime in $3$ different ways:
 * $11 = 0^2 + 11 = 2^2 + 7 = 3^2 + 2$


 * The $7$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$:
 * $11 = 3 + 8$


 * The $5$th Lucas number after $(2)$, $1$, $3$, $4$, $7$:
 * $11 = 4 + 7$


 * The $4$th Lucas prime after $2$, $3$, $7$.


 * A palindromic number whose square is also palindromic:


 * Cannot be represented by the sum of less than $6$ hexagonal numbers:
 * $11 = 6 + 1 + 1 + 1 + 1 + 1$


 * The $3$rd of the sequence of $n$ such that $p_n \# - 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $3$, $5$:
 * $p_{11} \# - 1 = 2309$


 * The $5$th of the sequence of $n$ such that $p_n \# + 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $2$, $3$, $5$, $7$:
 * $p_{11} \# + 1 = 2311$


 * The $4$th of the lucky numbers of Euler after $2$, $3$, $5$:
 * $n^2 + n + 11$ is prime for $0 \le n < 9$.


 * The $2$nd Thabit number after $(2)$, $5$, and $3$rd Thabit prime:
 * $11 = 3 \times 2^2 - 1$


 * The $2$nd positive integer $n$ after $5$ such that no factorial of an integer can end with $n$ zeroes.


 * The $6$th positive integer which is not the sum of $1$ or more distinct squares:
 * $2$, $3$, $6$, $7$, $8$, $11$, $\ldots$


 * The $8$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The $10$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$:
 * $11 = 11 \times 1 = 11 \times \left({1 \times 1}\right)$


 * The $4$th integer $m$ after $1$, $2$, $3$ such that $m! + 1$ is prime


 * The $1$st of $11$ primes of the form $2 x^2 + 11$:
 * $2 \times 0^2 + 11 = 11$


 * The $3$rd of $3$ primes of the form $2 x^2 + 3$:
 * $2 \times 2^2 + 3 = 11$

Also see

 * Divisibility by 11


 * 11 is Only Palindromic Prime with Even Number of Digits
 * Square of Small-Digit Palindromic Number is Palindromic