31 is Smallest Prime whose Reciprocal has Odd Period

Theorem
$31$ is the smallest prime number to have a decimal expansion of the reciprocal with an odd period greater than $1$:


 * $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

Proof
From Reciprocal of $31$:


 * $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

Counting the digits, it is seen that this has a period of recurrence of $15$, an odd integer.

The prime numbers less than $31$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$.

We investigate the reciprocal of each of these:

and it is seen that none has an odd period greater than $1$.

Hence the result.