Let a and b denote nonzero
elements of the ring of integers OK 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 (an − bn) if
p|(an − bn) and p ∤ (ak − bk) for k < n.
Definition 2. An integer n is called exceptionalfor {a,b} if (an − bn) 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 > n0(l) then n∉E(a,b), where
l = [K : Q] and n0 an effectively computable integer. In particular card
E(a,b) ≤ n0. 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.
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