317

Number

$317$ (three hundred and seventeen) is:

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 $2 x^2 + 29$:

$2 \times 12^2 + 29 = 317$ (Previous ... Next)

The prime nature of the repunit $R_{317}$ was discovered by Hugh Cowie Williams in $1978$.

John David Brillhart had previously mistakenly identified it as composite.

No further terms of this sequence are documented on $\mathsf{Pr} \infty \mathsf{fWiki}$.