Schinzel’s theorem

Definition 1.

Let $A$ and $B$ be integers such that $(A,B)=1$ with $AB\neq\pm 1$. A prime $p$ is called a primitive divisor of $A^{n}-B^{n}$ if $p$ divides $A^{n}-B^{n}$ but $A^{m}-B^{m}$ 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 $(A,B)=1$ and $A/B$ is not a root of unity. A prime ideal $\wp$ of $K$ is called a primitive divisor of $A^{n}-B^{n}$ if $\wp|A^{n}-B^{n}$ but $\wp\nmid A^{m}-B^{m}$ 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 $n_{0}$, depending only on the degree of the algebraic number $A/B$, such that $A^{n}-B^{n}$ has a primitive divisor for all $n>n_{0}$.

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

Corollary.

Let $A\neq 0,\pm 1$ be an integer. There exists a number $n_{0}$ such that $A^{n}-1$ has a primitive divisor for all $n>n_{0}$. 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 $n$.

References

• 1 A. Schinzel, Primitive divisors of the expression $A^{n}-B^{n}$ in algebraic number fields. Collection 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 SchinzelsTheorem 2013-03-22 12:03:42 2013-03-22 12:03:42 alozano (2414) alozano (2414) 10 alozano (2414) Theorem msc 20K01