# Composite Fermat number implies Poulet number

## Statement

Suppose  is a nonnegative integer. Let  be the  Fermat number:

.

Then:

.

In particular, if  is composite, then  is a Poulet number.

## Proof

Given: A nonnegative integer , .

To prove: .

Proof: For any nonnegative integer , . Thus, .

Now, for a Fermat number , the order of  modulo  is precisely . From the above, we conclude that the order of  modulo  divides , so .