# 11

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## Contents

## 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 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 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 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 $7$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:
- $5^{11} = 48 \, 828 \, 125$

- A palindromic number whose square is also palindromic:

- $11^2 = 121$

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

## Also see

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

### Previous in Sequence: $1$

### Previous in Sequence: $3$

*Previous ... Next*: Sequence of Integers whose Factorial plus 1 is Prime*Previous ... Next*: Palindromic Indices of Palindromic Triangular Numbers

### Previous in Sequence: $5$

*Previous ... Next*: Sequence of Prime Primorial minus 1*Previous ... Next*: Euler Lucky Number*Previous ... Next*: Numbers of Zeroes that Factorial does not end with*Previous ... Next*: Sophie Germain Prime*Previous ... Next*: Thabit Number*Previous ... Next*: Thabit Prime

### Previous in Sequence: $7$

*Previous ... Next*: Prime Number*Previous ... Next*: Twin Primes*Previous ... Next*: Permutable Prime*Previous ... Next*: Lucas Number*Previous ... Next*: Record Gaps between Twin Primes*Previous ... Next*: Safe Prime*Previous ... Next*: Lucas Prime*Previous ... Next*: Sequence of Prime Primorial plus 1*Previous ... Next*: Sequence of Integers whose Cube is Palindromic*Previous ... Next*: Palindromic Prime

### Previous in Sequence: $8$

### Previous in Sequence: $9$

*Previous ... Next*: Zuckerman Number

### Previous in Sequence: $10$

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation

### Next in Sequence: $23$ and above

*Next*: Sequence of Indices of Composite Mersenne Numbers*Next*: Prime Factors of One More than Power of 10*Next*: Repunit Prime

## 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

- $1760 = 11 \times 160$ yards equal $1$ (international) mile.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $11$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $11$

Categories:

- Work To Do
- Repunits/Examples
- Powers of 11/Examples
- Euler Lucky Numbers/Examples
- Sophie Germain Primes/Examples
- Thabit Numbers/Examples
- Thabit Primes/Examples
- Prime Numbers/Examples
- Twin Primes/Examples
- Permutable Primes/Examples
- Lucas Numbers/Examples
- Safe Primes/Examples
- Lucas Primes/Examples
- Palindromic Primes/Examples
- Ulam Numbers/Examples
- Zuckerman Numbers/Examples
- Repunit Primes/Examples
- Specific Numbers
- 11