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.

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 .