Composite Fermat number implies Poulet number
Suppose is a nonnegative integer. Let be the Fermat number:
In particular, if is composite, then is a Poulet number.
- Prime divisor of Fermat number is congruent to one modulo large power of two
- Mersenne number for prime or Poulet is prime or Poulet
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 .