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 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 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
Previous in Sequence: $1$
Previous in Sequence: $3$
- Previous ... Next: Square of Small-Digit Palindromic Number is Palindromic
- Previous ... Next: Sequence of Integers whose Factorial plus 1 is Prime
- Previous ... Next: Sequence of Smallest Numbers whose Reciprocal has Period n
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- 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 ... Next: Solutions of Ramanujan-Nagell Equation
Previous in Sequence: $7$
- Previous ... Next: Prime Number
- Previous ... Next: Twin Primes
- Previous ... Next: Permutable Prime
- Previous ... Next: Integer Partition
- Previous ... Next: Lucas Number
- Previous ... Next: Record Gaps between Twin Primes
- Previous ... Next: Minimal Prime
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- 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 ... Next: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares
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: $13$ 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:
- Repunits/Examples
- Powers of 11/Examples
- Unique Period Primes/Examples
- Euler Lucky Numbers/Examples
- Sophie Germain Primes/Examples
- Thabit Numbers/Examples
- Thabit Primes/Examples
- Prime Numbers/Examples
- Twin Primes/Examples
- Permutable Primes/Examples
- Integer Partitions/Examples
- Lucas Numbers/Examples
- Minimal Primes/Examples
- Safe Primes/Examples
- Lucas Primes/Examples
- Palindromic Primes/Examples
- Ulam Numbers/Examples
- Zuckerman Numbers/Examples
- Repunit Primes/Examples
- Specific Numbers
- 11