# 11

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## 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 $2 x^2 + 11$:
$2 \times 0^2 + 11 = 11$ (Next)

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 $3$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 $2 x^2 + 3$:
$2 \times 2^2 + 3 = 11$ (Previous)

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:
$11 = 6 + 1 + 1 + 1 + 1 + 1$

## Historical Note

Some of the archaic imperial units of length appear as divisors and multiples of other such units with a multiplicity of $11$, for example:

$4$ rods, poles or perches equal $22$ yards equal $1$ chain
$220 = 11 \times 20$ yards equal $1$ furlong
$1760 = 11 \times 160$ yards equal $1$ (international) mile.