Displaying similar documents to “Prime divisors of the Lagarias sequence”

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.

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...

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.

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.

Short remark on Fibonacci-Wieferich primes

Jiří Klaška (2007)

Acta Mathematica Universitatis Ostraviensis

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This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime p is Fibonacci-Wieferich is equal to 1 / p . According to our computational results and some theoretical consideratons, another form of probability...