Composite Fermat number: Difference between revisions

Latest revision as of 20:01, 20 April 2009

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 $2$ .

@@ Line 2: / Line 2: @@
 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|[[Prime divisor of Fermat number is congruent to one modulo large power of two]]}}
+If <math>k \ge 3</math>, and <math>F_k</math> is a composite Fermat number, then <math>2^{k+2}</math> divides <math>p - 1</math> for any prime divisor <math>p</math> of <math>F_k</math>. As a corollary, <math>2^{k+2}</math> divides <math>m-1</math> for ''any'' divisor <math>m</math> of <math>F_k</math>.
 ==Relation with other properties==