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 of the $4$th pair of twin primes, with $17$


 * 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 $2$nd Keith number:
 * $1, 9, 10, 19, \ldots$


 * 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 $3$rd prime number whose period is of maximum length:
 * $\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$


 * The $3$rd of $11$ primes of the form $2 x^2 + 11$:
 * $2 \times 2^2 + 11 = 19$


 * 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 $16$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.


 * 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 $12$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.

Also see

 * Period of Reciprocal of 19 is of Maximal Length
 * Hilbert-Waring Theorem for $4$th Powers