An extension of the theorem on primitive divisors in algebraic number fields

Abstract:

The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers, $(A,B) = 1, AB \ne 0$, A/B not a root of unity, and ${\zeta _k}$ a primitive root of unity of order k. For all sufficiently large n, the number ${A^n} - {\zeta _k}{B^n}$ has a prime ideal factor that does not divide ${A^m} - \zeta _k^j{B^m}$ for arbitrary $m < n$ and $j < k$.