11

Number
$11$ (eleven) is:


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


 * The only palindromic prime with an even number of digits


 * The $1$st power of $11$ after the zeroth $1$:
 * $11 = 11^1$


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


 * The $1$st repunit prime


 * The $1$st prime $p$ such that the Mersenne number $2^p - 1$ is composite:
 * $2^{11} - 1 = 2047 = 23 \times 89$


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


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


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


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


 * The smallest positive integer the decimal expansion of whose reciprocal has a period of $2$:
 * $\dfrac 1 {11} = 0 \cdotp \dot 0 \dot 9$


 * The $3$rd prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $3$, $5$:
 * $11 \# - 1 = 2 \times 3 \times 5 \times 11 - 1 = 2309$


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


 * The $3$rd safe prime after $5$, $7$:
 * $11 = 2 \times 5 + 1$


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


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


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


 * The $4$th positive integer after $1$, $2$, $7$ whose cube is palindromic:
 * $11^3 = 1331$


 * The $5$th palindromic integer after $0$, $1$, $2$, $3$ which is the index of a palindromic triangular number
 * $T_{11} = 66$


 * The $5$th palindromic integer after $0$, $1$, $2$, $3$ whose square is also palindromic integer
 * $11^2 = 121$


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


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


 * The $5$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $0$, $1$, $2$, $3$, $11$


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


 * The $5$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $2$, $3$, $5$, $7$:
 * $11 \# + 1 = 2 \times 3 \times 5 \times 11 - 1 = 2311$


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


 * The $6$th odd positive integer after $1$, $3$, $5$, $7$, $9$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime


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


 * 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 $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$:
 * $11 = 11 \times 1 = 11 \times \paren {1 \times 1}$


 * The $11$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$ such that $5^n$ contains no zero in its decimal representation:
 * $5^{11} = 48 \, 828 \, 125$


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

Also see

 * Divisibility by 11


 * 11 is Only Palindromic Prime with Even Number of Digits