On the prime ideal divisors of (aⁿ −bⁿ)

Edward Grossman

Vol. 54 (1974), No. 2, 73–83

DOI: 10.2140/pjm.1974.54.73

Abstract

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 (aⁿ − bⁿ) if p|(aⁿ − bⁿ) and p ∤ (a^k − b^k) 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.

Mathematical Subject Classification

Primary: 12A45

Secondary: 10A20

Milestones

Received: 23 July 1973

Published: 1 October 1974