Aurifeuillian factorization

The Cunningham project seeks to factor numbers of the form $b^{n}\pm 1$ with $b=2,3,\dots$ small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing $b^{n}$ by a polynomial in such a way that polynomial factorization is possible. For example, by substituting $y=2^{k}$ into the polynomial factorization $(2y^{2})^{2}+1=(2y^{2}-2y+1)(2y^{2}+2y+1)$ we can partially factor $2^{4k+2}+1$. In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified all numbers that can be factored by such identities and we prove this if one accepts our definition of what ``such an identity'' is. We then develop our theme to similarly factor $f(b^{n})$ for any given polynomial $f$, using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.