Reciprocal of 83 has Prime Period

Theorem

The decimal expansion of the reciprocal of $83$ has a prime period, that is $41$:

$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$

Proof

From Reciprocal of $83$:

$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$

Counting the digits, it is seen that this has a period of recurrence of $41$.

Indeed, $41$ is the $13$th prime number.

$\blacksquare$