# 317

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## Number

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

- The $66$th prime number

- 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 $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$:

## Historical Note

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.

## Also see

*Previous ... Next*: Index of Repunit Prime*Previous ... Next*: Sequence of Prime Primorial minus 1*Previous ... Next*: Prime Gaps of 14*Previous ... Next*: Prime Number*Previous ... Next*: Two-Sided Prime*Previous*: Sequences of 3 Consecutive Integers with Rising Phi

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $11,111,111, \ldots 111,111$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $11,111,111, \ldots 111,111$