Composite Fermat number

Definition

A composite Fermat number is a Fermat number that is also a composite number, i.e., it is not a Fermat prime.

Facts

Constraints on prime divisors

Further information: Prime divisor of Fermat number is congruent to one modulo large power of two

If $k\geq 3$, and $F_{k}$ is a composite Fermat number, then $2^{k+2}$ divides $p-1$ for any prime divisor $p$ of $F_{k}$. As a corollary, $2^{k+2}$ divides $m-1$ for any divisor $m$ of $F_{k}$.

Relation with other properties

Weaker properties

• Composite Proth number
• Poulet number: It is a Fermat pseudoprime to the base $2$.
• Strong pseudoprime to base $2$.