Long Period Prime/Examples/7

Theorem

$7$ is the smallest long period prime:

$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

$\dfrac 1 7 = 0 \cdotp \dot 14285 \, \dot 7$

The reciprocals of $1$, $2$, $4$ and $5$ do not recur:

$\ds \frac 1 1$

$=$

$\ds 1$

$\ds \frac 1 2$

$=$

$\ds 0 \cdotp 5$

$\ds \frac 1 4$

$=$

$\ds 0 \cdotp 25$

$\ds \frac 1 5$

$=$

$\ds 0 \cdotp 2$

while those of $3$ and $6$ do recur, but with the non-maximum period of $1$:

$\ds \frac 1 3$

$=$

$\ds 0 \cdotp \dot 3$

$\ds \frac 1 6$

$=$

$\ds 0 \cdotp 1 \dot 6$

$\blacksquare$