# Composite Fermat number implies Poulet number

From Number

## Statement

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

.

Then:

.

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

## Related facts

- Prime divisor of Fermat number is congruent to one modulo large power of two
- Mersenne number for prime or Poulet is prime or Poulet

## 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 .