Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order

Theorem
Let $D$ be a deck of $2 m$ cards.

Let $D$ be given a sequence of modified perfect faro shuffles.

Then the cards of $D$ will return to their original order after $n$ such shuffles, where:
 * $2^n \equiv 1 \pmod {2 m + 1}$

Proof
From Position of Card after n Modified Perfect Faro Shuffles, after $n$ shuffles a card in position $x$ will be in position $2^n x \pmod {m + 1}$.

So for all $2 m$ cards in $D$, we need to find $n$ such that:
 * $2^n x \equiv x \pmod {2 m + 1}$

Because $2 m + 1$ is odd, we have:
 * $\gcd \set {2, 2 m + 1}$

and so from Cancellability of Congruences:


 * $2^n \equiv 1 \pmod {2 m + 1}$