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 $n$.
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 $n$.
References
- 1 A. Schinzel, Primitive divisors of the expression ${A}^{n}\mathrm{-}{B}^{n}$ 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 |
---|---|
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 |