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 $2$nd unique period prime after $3$: its period is $2$:
::$\dfrac 1 {11} = 0 \cdotp \dot 0 \dot 9$
: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 smaller 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 $4$th positive integer solution after $1$, $3$, $5$ to the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ for integral $n$:
::$11^2 + 7 = 32 = 2^7$
: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 $5$th minimal prime base $10$ after $2$, $3$, $5$, $7$
: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 number of integer partitions for $6$:
::$\map p 6 = 11$
: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
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=== Next in Sequence: $13$ and above ===
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== Historical Note ==
== Sources ==
* : $11$
* : $11$
Category:Specific Numbers
Category:11