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 $2$, $3$, $5$, $7$, $23$, $37$, $53$, $73$, $313$:
- The $19$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$, $313$
- The $24$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$, $283$, $313$
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: Prime Gaps of 14
- Previous ... Next: Prime Number
- Previous ... Next: Left-Truncatable Prime
- Previous ... Next: Right-Truncatable Prime
- Previous ... Next: Two-Sided Prime
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): $11,111,111, \ldots 111,111$