Schinzel’s theorem

Definition 1.

Let A and B be integers such that (A,B)=1 with AB±1. A prime p is called a primitive divisorMathworldPlanetmathPlanetmath of An-Bn if p divides An-Bn but Am-Bm is not divisible by p for all positive integers m that are less than n.

Or, more generally:

Definition 2.

Let A and B be algebraic integersMathworldPlanetmath in a number fieldMathworldPlanetmath K such that (A,B)=1 and A/B is not a root of unityMathworldPlanetmath. A prime idealMathworldPlanetmathPlanetmath of K is called a primitive divisor of An-Bn if |An-Bn but Am-Bm for all positive integers m that are less than n.

The following theorem is due to A. Schinzel (see [1]):


Let A and B be as before. There is an effectively computable constant n0, depending only on the degree of the algebraic numberMathworldPlanetmath A/B, such that An-Bn has a primitive divisor for all n>n0.

By putting B=1 we obtain the following corollary:


Let A0,±1 be an integer. There exists a number n0 such that An-1 has a primitive divisor for all n>n0. In particular, for all but finitely many integers n, there is a prime p such that the multiplicative orderMathworldPlanetmath of A modulo p is exactly n.


  • 1 A. Schinzel, Primitive divisors of the expression An-Bn in algebraic number fields. CollectionMathworldPlanetmath of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. J. Reine Angew. Math. 268/269 (1974), 27–33.
Title Schinzel’s theorem
Canonical name SchinzelsTheorem
Date of creation 2013-03-22 12:03:42
Last modified on 2013-03-22 12:03:42
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Theorem
Classification msc 20K01