In this paper, we present a class
of new  Aurifeuillian factorization of $M^n\pm 1$,e.i.:
 Let positive integer $m\equiv \epsilon (\mathrm{ mod}
4),\epsilon=1,-1,n=mk$, where $ n\equiv 1(\mathrm{ mod} 2),k \in Z^+$. 
If $M$ is a multiple of $ m$ and $\frac{M}{m}$ is a square, then
$\Phi_n(\epsilon M)=(\Phi_n(\epsilon M),\Delta_{\epsilon,1})(\Phi_n(\epsilon
M),\Delta_{\epsilon,2})$, and $
(\Phi_n(\epsilon M),\Delta_{\epsilon,1})=(\Phi_n(\epsilon M),\mathrm{
Norm}_{Q(\eta_m)/Q}(\sqrt{\epsilon M}^k-\eta_m))$ and
$(\Phi_n(\epsilon M),\Delta_{\epsilon,2})=(\Phi_n(M),\mathrm{
Norm}_{Q(\eta_m)/Q}(\sqrt{\epsilon M}^k+\eta_m))$, where
$\Delta_{\epsilon,r}=mM^{k\frac{m+1}{2}}+(-1)^r(\frac{2}{m})\sqrt{mM}
M^{\frac{k-1}{2}}$
$\sum\limits_{\stackrel{c=1}{(c,m)=1}}^m(\frac{c}{m}){(\epsilon M)}^{kc}
\;\;\; r=1,2. 
$\,\, Finally we give an example about the Aurifeuillian factorization
 of a very large cyclotomic number with 362 digits.