Displaying similar documents to “The correction factor in Artin's primitive root conjecture”

Prime divisors of the Lagarias sequence

Pieter Moree, Peter Stevenhagen (2001)

Journal de théorie des nombres de Bordeaux

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We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence x n n = 1 ∞ defined by the linear recurrence x n + 1 = x n + x n - 1 and the initial values x 0 = 3 and x 1 = 1 . This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.

Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Hans Roskam (2001)

Journal de théorie des nombres de Bordeaux

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Let S be a linear integer recurrent sequence of order k ≥ 3 , and define P S as the set of primes that divide at least one term of S . We give a heuristic approach to the problem whether P S has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that P S has positive lower density for “generic” sequences S . Some numerical examples are included.

Artin's primitive root conjecture for quadratic fields

Hans Roskam (2002)

Journal de théorie des nombres de Bordeaux

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Fix an element α in a quadratic field K . Define S as the set of rational primes p , for which α has maximal order modulo p . Under the assumption of the generalized Riemann hypothesis, we show that S has a density. Moreover, we give necessary and sufficient conditions for the density of S to be positive.

Frobenius distributions for real quadratic orders

Peter Stevenhagen (1995)

Journal de théorie des nombres de Bordeaux

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We present a density result for the norm of the fundamental unit in a real quadratic order that follows from an equidistribution assumption for the infinite Frobenius elements in the class groups of these orders.