Let a and b denote nonzero
elements of the ring of integers O_{K} of an algebraic number field K, such that
ab^{−1} is not a root of unity and the principal ideals (a) and (b) are relatively
prime.
Definition 1. A prime ideal p is called a primitive prime divisor of (a^{n} − b^{n}) if
p(a^{n} − b^{n}) and p ∤ (a^{k} − b^{k}) for k < n.
Definition 2. An integer n is called exceptionalfor {a,b} if (a^{n} − b^{n}) has no
primitive prime divisors.
The set of integers exceptional for {a,b} is denoted by E(a,b). Using recent deep
results of Baker, Schinzel [4] has proved that if n > n_{0}(l) then n∉E(a,b), where
l = [K : Q] and n_{0} an effectively computable integer. In particular card
E(a,b) ≤ n_{0}. In this paper, using only elementary methods, upper bounds are
obtained for card {n ∈ E(a,b) : n ≤ x} which are independent of a and
b.
