A question about Aurifeuillian factorizations (Q1909440)

Factorisations of numbers of the form \(b^n \pm 1\) reduce to that of cyclotomic polynomials, and some such numbers possess Aurifeuillian factorisations. \textit{S. Hahn} [Math. Jap. 39, No. 3, 501--502 (1994; Zbl 0819.11003)] linked such factorisations to Gauss sums and norms in cyclotomic fields. There remained the problem of whether such a factorisation is always the same as that from such considerations associated with a prime \(p\equiv 1\pmod 4\). Here the authors provide the affirmative answer, namely that the two factorisations are in fact always the same. They also deal with the case \(p\equiv 3\pmod 4\).