The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon and its generalizations. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (about the zeros of the \hbox{Dirichlet} $L$-function), we can characterize exactly those moduli and residue classes for which the bias is present. We also give results of numerical investigations on the prevalence of the bias for several moduli. Finally, we briefly discuss generalizations of the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces.

We describe an implementation of the generalized Lagrange algorithm for computing units in algebraic number fields [Buchmann 1987a], together with extensive experimental data of the algorithm's application (to all totally real quartic fields of discriminant below 60000). We also present an improved algorithm, with related experimental data.

The purpose of this paper is to show that elliptic diophantine equations cannot always be solved--in the most practical sense--by the Thue approach, that is, by solving each of the finitely many corresponding Thue equations of degree 4. After a brief general discussion, which is necessarily of a heuristic nature, to substantiate our claim, we consider the elliptic equation associated with the Ochoa curve. An explicit computational explanation as to the reasons for the failure of the Thue approach in this case is followed by a complete solution of the standard Weierstraß equation of this elliptic curve by a method which makes use of a recent lower bound for linear forms in elliptic logarithms.

The study of resonances in systems such as periodically forced oscillators has traditionally focused on understanding the regions in the parameter plane where these resonances occur. Resonance regions can also be viewed as projections to the parameter plane of resonance surfaces in the four-dimensional Cartesian product of the state space with the parameter space. This paper reports on a computer study of resonance surfaces for a particular family and illustrates some advantages of viewing resonance regions in this light.

We describe a new algorithm, based on sieving procedures, for determining the minimal index and all elements with minimal index in a class of totally real quartic fields with Galois group $D_8$. It is not universally applicable, but its applicability is easily checked for any particular example, and it is very fast when applicable. We include several tables demonstrating the potential of the method. (A more general approach for quartic fields, described in [Gaál et al.], requires much more computation time for each field.)

Finally, we present a family of totally real quartic fields with Galois group $D_8$ and having minimal index 1 (that is, a power integral basis).

Baumslag defined a family of groups that are of interest because they closely resemble free groups, yet are not free. It was known that each group in this family has the same lower central series of quotients and the same first two terms in the derived series of quotients as does the free group $F$ on two generators.

We have verified that there are different isomorphism types among the groups in the family, and that the third terms in the derived series of quotients are often distinct from that of $F$. Our basic technique is to count the number of homomorphisms from the groups of interest to a target group.