Midy's Theorem

Theorem
Let $p$ be a prime.

Let $a \in \set {1, 2, \ldots, p - 1}$ and $b > 1$ be integers.

Let $N$ be the recurring part of the expansion of $\dfrac a p$ in base $b$.

Let $\alpha$ be the period of recurrence of $N$.

Let $\alpha = c k$ for (strictly) positive integers $c > 1$ and $k$.

Then $N$ is divisible by $b^k - 1$.

Moreover, let $N = \ds \sum_{i \mathop = 1}^c N_i \paren {b^k}^i$ for $N_i \in \set {0, 1, \ldots b^k - 1}$.

Then:
 * $\ds \sum_{i \mathop = 1}^c N_i = r \paren {b^k - 1}$ for some $r \in \set {1, 2, \ldots, c - 1}$

$N$ is divisible by $b^k - 1$
By definition of recurrence, we have:


 * $\dfrac a p b^\alpha = N + \dfrac a p$

Moreover, by definition of period of recurrence, $\alpha$ is the smallest positive integer for which this is true.

Rearranging, we obtain:


 * $\dfrac a p \paren {b^\alpha - 1} = N$

In particular, $a \paren {b^\alpha - 1}$ is divisible by $p$.

As $p$ is prime:
 * $p \divides a$ or $p \divides \paren {b^\alpha - 1}$

The former is false because $0 < a < p$.

Therefore, we must have $p \divides \paren {b^\alpha - 1}$.

If $\alpha = c k$ with $c > 1$, then $b^\alpha - 1 = \paren {b^k - 1} m$ for some integer $m$.

As $p$ is prime:
 * $p \divides b^k - 1$ or $p \divides m$

The former is false because $0 < k < \alpha$ and $\alpha$ is the smallest positive integer for which this is true.

Therefore, we must have $p \divides m$.

This implies that $a \dfrac m p$ is an integer.

We then have:


 * $a \dfrac m p \paren {b^k - 1} = N$

By definition, $N$ is divisible by $b^k - 1$.

The Sum of the $N_i$ is divisible by $b^k - 1$
Calculating modulo $b^k - 1$, we have:

Therefore, we have:


 * $\ds \sum_{k \mathop = 1}^c N_i = r \paren {b^k - 1}$

Notice further that:
 * $\ds 0 < \sum_{k \mathop = 1}^c N_i < \sum_{k \mathop = 1}^c \paren {b^k - 1}$

The latter strict inequality follows because otherwise, we would have $N_i = b^k - 1$, meaning $\alpha = 1$.

Therefore, we must have $1 < r < c-1$.

Also see

 * Definition:Repdigit Number
 * Number times Recurring Part of Reciprocal gives 9-Repdigit