Displaying similar documents to “Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields”

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.

The correction factor in Artin's primitive root conjecture

Peter Stevenhagen (2003)

Journal de théorie des nombres de Bordeaux

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In 1927, E. Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root modulo p . After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first...