37

Number
$37$ (thirty-seven) is:


 * The $12$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$


 * Every positive integer can be expressed as the sum of at most $37$ positive $5$th powers


 * The $2$nd integer which has a reciprocal whose period is $3$:
 * $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$


 * The $3$rd unique period prime: its period is $3$:
 * $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$


 * The $3$rd of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 2^2 + 29 = 37$


 * The $4$th centered hexagonal number after $1$, $7$, $19$:
 * $37 = 1 + 6 + 12 + 18 = 4^3 - 3^3$


 * The $4$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$:
 * $37 = 6^2 + 1$


 * The $4$th emirp after $13$, $17$, $31$


 * The $4$th prime $p$ after $11$, $23$, $29$ such that the Mersenne number $2^p - 1$ is composite


 * The $5$th of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 4^2 + 5 = 37$


 * The $6$th two-sided prime after $2$, $3$, $5$, $7$, $23$:
 * $37$, $3$, $7$ are prime


 * The $7$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $0$, $1$, $2$, $3$, $11$, $27$, $37$


 * The $8$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$


 * The $8$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$


 * The $9$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$


 * The $11$th lucky number:
 * $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $\ldots$


 * The $19$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $37$, $\ldots$


 * The $23$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$ which cannot be expressed as the sum of distinct pentagonal numbers


 * The $27$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $27$, $30$, $31$, $32$, $35$, $36$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$


 * The $27$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{37} = 137 \, 438 \, 953 \, 472$

Also see

 * Period of Reciprocal of 37 has Length 3
 * Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples
 * Hilbert-Waring Theorem for $5$th Powers