Vol. 54, No. 2, 1974

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On the prime ideal divisors of (an bn)

Edward Grossman

Vol. 54 (1974), No. 2, 73–83

Let a and b denote nonzero elements of the ring of integers OK of an algebraic number field K, such that ab1 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 nE(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.

Mathematical Subject Classification
Primary: 12A45
Secondary: 10A20
Received: 23 July 1973
Published: 1 October 1974
Edward Grossman