### A note on factorization of the Fermat numbers and their factors of the form $3h{2}^{n}+1$

We show that any factorization of any composite Fermat number ${F}_{m}={{2}^{2}}^{m}+1$ into two nontrivial factors can be expressed in the form ${F}_{m}=(k{2}^{n}+1)(\ell {2}^{n}+1)$ for some odd $k$ and $\ell ,k\ge 3,\ell \ge 3$, and integer $n\ge m+2,3n<{2}^{m}$. We prove that the greatest common divisor of $k$ and $\ell $ is 1, $k+\ell \equiv 0\phantom{\rule{4pt}{0ex}}mod{2}^{n},\phantom{\rule{4pt}{0ex}}max(k,\ell )\ge {F}_{m-2}$, and either $3|k$ or $3|\ell $, i.e., $3h{2}^{m+2}+1|{F}_{m}$ for an integer $h\ge 1$. Factorizations of ${F}_{m}$ into more than two factors are investigated as well. In particular, we prove that if ${F}_{m}={(k{2}^{n}+1)}^{2}(\ell {2}^{j}+1)$ then $j=n+1,3|\ell $ and $5|\ell $.