The formula \(a^n-b^n=\prod_{d\mid n}\Phi_d(a,b)\), where the \(\Phi_d(X,Y)\) are cyclotomic polynomials, enables one to reduce the factorization of \(a^n-b^n\) \((a,b\in\mathbb Z,\;n>2)\) to that of \(\Phi_d(a,b)\). There are factorizations of \(\Phi_d(a,b)\), called Aurifeuillian factorizations, which stem from certain types of identities satisfied by the polynomials \(\Phi_d(X,Y)\). The author gives a simple argument for the existence of these identities and infers properties of the (Aurifeuillian) polynomials involved. He applies this result to the splitting of \(\Phi_d(a,b)\), studies the coprimality of the factors and discusses the relationship between various factorizations of one and the same number, proving results suggested by J. Brillhart et al. [Factorizations of \(b^n\pm 1,\dots\), Contemp. Math. 22, 178 p. (1983; Zbl 0527.10001)]. The final section furnishes a fast way of computing the Aurifeuillian polynomials by means of the Euclidean algorithm.
As remarked by the author, the first result mentioned above (identities for \(\Phi_d(X,Y))\) is due to A. Schinzel [Proc. Camb. Philos. Soc. 58, 555–562 (1962; Zbl 0114.02605)].