19

== Number ==
$19$ (nineteen) is:
:The $8$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$
:The index of the $2$nd repunit prime after $R_2$:
::$R_{19} = 1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$
:The $2$nd prime number after $5$ of the form $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
::$19 = 4! - 3! + 2! - 1!$
:The $2$nd Keith number after $14$:
::$1$, $9$, $10$, $19$, $\ldots$
:The lower end of the $3$rd record-breaking gap between twin primes:
::$29 - 19 = 10$
:The $3$rd centered hexagonal number after $1$, $7$:
::$19 = 1 + 6 + 12 = 3^3 - 2^3$
:The $3$rd long period prime after $7$, $17$:
::$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
:The $3$rd after $2$, $3$ of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
::$\dfrac {19^{19} - 1} {19 - 1} = 109 \, 912 \, 203 \, 092 \, 239 \, 643 \, 840 \, 221$
:The $3$rd integer after $7$, $13$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$19^2 = 361$; $36 = 6^2$, $1 = 1^2$
:The $3$rd of $11$ primes of the form $2 x^2 + 11$:
::$2 \times 2^2 + 11 = 19$ ()
:The larger of the $4$th pair of twin primes, with $17$
:The $5$th happy number after $1$, $7$, $10$, $13$:
::$19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $5$th positive integer after $6$, $9$, $12$, $18$ whose cube can be expressed as the sum of $3$ positive cube numbers:
::$19^3 = 3^3 + 10^3 + 18^3$
:The $6$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$
:The index of the $7$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$:
::$M_{19} = 2^{19} - 1 = 524 \, 287$
:The index of the $8$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$ which asserted to be prime
:The $9$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime
:The $10$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $\ldots$
:The $10$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $\ldots$
:The $12$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $15$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $18$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $16$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$ such that $2^n$ contains no zero in its decimal representation:
::$2^{19} = 524 \, 288$
:Every positive integer can be expressed as the sum of at most $19$ $4$th powers.
== Also see ==
* Reciprocal of 19
* Hilbert-Waring Theorem for $4$th Powers
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== Sources ==
* : $19$
* : $1,111,111,111,111,111,111$
* : $19$
* : $1,111,111,111,111,111,111$
Category:Specific Numbers
Category:19