Schinzel’s theorem
Definition 1.
Let and be integers such that with . A prime is called a primitive divisor of if divides but is not divisible by for all positive integers that are less than .
Or, more generally:
Definition 2.
Let and be algebraic integers in a number field
such that and is not a root of unity
. A prime ideal
of is called a primitive divisor of if but for all positive integers that are less than .
The following theorem is due to A. Schinzel (see [1]):
Theorem.
Let and be as before. There is an effectively computable constant , depending only on the degree of the algebraic number , such that has a primitive divisor for all .
By putting we obtain the following corollary:
Corollary.
Let be an integer. There exists a number such that has a primitive divisor for all . In particular, for all but finitely many integers , there is a prime such that the multiplicative order of modulo is exactly .
References
- 1 A. Schinzel, Primitive divisors of the expression 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.