317

Number
$317$ (three hundred and seventeen) is:


 * The $66$th prime number


 * The smaller of the $3$rd pair of primes whose prime gap is $14$:
 * $331 - 317 = 14$


 * The $3$rd of the $3$rd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ and $\tuple {165, 166, 167}$ that have Euler $\phi$ values which are strictly increasing:
 * $\map \phi {315} = 144$, $\map \phi {316} = 156$, $\map \phi {317} = 316$


 * The index of the $4$th repunit prime after $R_2$, $R_{19}$, $R_{23}$


 * The $7$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
 * $3$, $5$, $11$, $13$, $41$, $89$, $317$


 * The $10$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$:
 * $317$ is prime; $31$, $3$ are prime; $17$, $7$ are prime


 * The $13$th of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 12^2 + 29 = 317$