Schinzel’s theorem

Definition 1.

Let $A$ and $B$ be integers such that $\mathrm{(}A\mathrm{,}B\mathrm{)}\mathrm{=}\mathrm{1}$ with $A\mathit{}B\mathrm{\ne }\mathrm{\pm }\mathrm{1}$. A prime $p$ is called a primitive divisor of $A^n\mathrm{-}{B}^n$ if $p$ divides $A^n\mathrm{-}{B}^n$ but $A^m\mathrm{-}{B}^m$ 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 integers in a number field $K$ such that $\mathrm{(}A\mathrm{,}B\mathrm{)}\mathrm{=}\mathrm{1}$ and $A\mathrm{/}B$ is not a root of unity. A prime ideal $\mathrm{\wp }$ of $K$ is called a primitive divisor of $A^n\mathrm{-}{B}^n$ if $\mathrm{\wp }\mathrm{|}{A}^n\mathrm{-}{B}^n$ but $\mathrm{\wp }\mathrm{\nmid }{A}^m\mathrm{-}{B}^m$ for all positive integers $m$ that are less than $n$.

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

Theorem.

Let $A$ and $B$ be as before. There is an effectively computable constant $n_{\mathrm{0}}$, depending only on the degree of the algebraic number $A\mathrm{/}B$, such that $A^n\mathrm{-}{B}^n$ has a primitive divisor for all $n\mathrm{>}{n}_{\mathrm{0}}$.

By putting $B=1$ we obtain the following corollary:

Corollary.

Let $A\mathrm{\ne }\mathrm{0}\mathrm{,}\mathrm{\pm }\mathrm{1}$ be an integer. There exists a number $n_{\mathrm{0}}$ such that $A^n\mathrm{-}\mathrm{1}$ has a primitive divisor for all $n\mathrm{>}{n}_{\mathrm{0}}$. In particular, for all but finitely many integers $n$, there is a prime $p$ such that the multiplicative order of $A$ modulo $p$ is exactly $n$.

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