Here is an expository write-up of this post.

The number is interesting because of the following property

and . The last equation means that

So multiplying by simply permutes the digits of . As elements of these permutations correspond to (using cycle notation)

, , , , ,

respectively. We can read off the following from this list.

- Since and are full-length cycles, and are primitive roots modulo .
- and have order modulo , while has order .

In general, if is an odd prime and is a positive integer with order modulo , then will have period of length in base . This follows simply from observing that if then

.

Now let be a primitive root modulo . We shall work in base . Then will have period of length . Since the periods of are , all of length , which are all powers of some permutation of the digits of , and no two of which are congruent modulo , it follows that the digits of are all distinct. So we’ve shown the following.

**Theorem. **If is an odd prime and is a primitive root modulo , then the base expansion of has period of length with distinct digits.

Taking , gives our previous example. So the prime in base is indeed very special, since it is the only prime less than for which is a primitive root.