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Frobenius distributions for real quadratic orders

Journal de théorie des nombres de Bordeaux

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.

The correction factor in Artin's primitive root conjecture

Journal de théorie des nombres de Bordeaux

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 is of historical...

Prime divisors of the Lagarias sequence

Journal de théorie des nombres de Bordeaux

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 ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ 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.

On the computation of quadratic $2$-class groups

Journal de théorie des nombres de Bordeaux

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer $D\equiv 0,1$ mod $4$ and the factorization of $D$, computes the structure of the $2$-Sylow subgroup of the class group of the quadratic order of discriminant $D$ in random polynomial time in $log\left|D\right|$.

The number of real quadratic fields having units of negative norm.

Experimental Mathematics

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