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We find all the maximal admissible connected sets of Gaussian primes: there are 52 of them. Our catalog corrects some errors in the literature. We also describe a totally automated procedure to determine the heuristic estimates for how often various patterns, in either the integers or Gaussian integers, occur in the primes. This heuristic requires a generalization of a classical formula of Mertens to the Gaussian integers, which we derive from a formula of Uchiyama regarding an Euler product that involves only primes congruent to 1 (mod 4).
We propose a theory to explain random behavior for the digits in the expansions of fundamental mathematical constants. At the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base-2 normality---namely bit randomness in a specific technical sense---for a collection of celebrated constants, including π, log 2, ζ(3), and others. Also on the hypothesis, the number ζ(5) is either rational or normal to base 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number generators.
In a well-known paper, Cohen and Lenstra gave conjectures on class groups of number fields. We give here similar conjectures for Tate-Shafarevitch groups of elliptic curves defined over ℚ. For such groups (if they are finite), there exists a nondegenerate, alternating, bilinear pairing. We give some properties of such groups and then formulate heuristics which allow us to give precise conjectures.
We describe a new algorithm for direct condensation, which is a tool in computational representation theory. The crucial point for this is the enumeration of very large orbits for a group acting on some set. We present a variation of the standard orbit enumeration algorithm that reduces the amount of storage needed and behaves well under parallelization. For the special case of matrices acting on a finite vector space an efficient implementation is described. This allows us to use condensation methods for considerably larger permutation representations than could be handled before.
We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible three-manifolds of complexity up to 9. More interestingly, we have actually been able to give a name to each such manifold, by recognizing its canonical decomposition into Seifert fibered spaces and hyperbolic manifolds.
The algorithm relies on the extension of Matveev's theory of complexity to the case of manifolds bounded by suitably marked tori, and on the notion of assembling of two such manifolds. We show that every manifold is an assembling of manifolds which cannot be further disassembled, and we prove that there are surprisingly few such manifolds up to complexity 9.
Our most interesting experimental discovery is that there are 4 closed hyperbolic manifolds having complexity 9, and they are the 4 closed hyperbolic manifolds of least known volume. All other manifolds having complexity at most 9 are graph manifolds.
We study the Artin L-function L(s, χ) associated to the unique character χ of degree 2 in quaternion fields of degree 8. We first explain how to find examples of quaternion octic fields with not too large a discriminant. We then develop a method using a quick compuation of the order n*#967; of the zero of L(s, χ) at the point s = 1/2. In all our calculations, we find that nχ only depends on the sign of the root number W(χ); indeed nχ = 0 when W (χ) = -1. Finally we give some estimates on nχ and low zeros of L(s, χ) on the critical line in terms fo the Artin conductor *#402;χ of the character χ.
The Teichmüller space of once-punctured tori can be realized as the upper half-plane ℍ, or via the Maskit embedding as a proper subset of ℍ. We construct and approximate the explicit biholomorphic map from Maskit's embedding to ℍ. This map involves the integration of an abelian differential constructed using an infinite sum over the elements of a Kleinian group. We approximate this sum and thereby find the locations of the square torus and the hexagonal torus in Maskit's embedding, and we show that the biholomorphism does not send vertical pleating rays in Maskit's embedding to vertical lines in ℍ.
We study the distribution of prime numbers that have a given number of one bits in their binary representation, and of those that have a given number of zero bits. We consider basic questions such as whether there are infinitely many of them, and explain their distribution in residue classes modulo small primes. We prove the unexpected result that, for m ≥ 3, there is no prime number with precisely 2m bits, exactly two of which are zero bits.
One essential part of Elkies' algorithm for computing the group order of an elliptic curve defined over a finite field is the determination of the eigenvalue of the Frobenius endomorphism. Here we compare form a practical point of view several strategies for this search: the use of rational functions, the use of division polynomials, the babystep-giantstep method, and a new modification of this method that avoids the need for two fast exponentiations.
We study the possible link between the dynamics of a certain family of circle maps and the caustics of their iterates. The maps are defined by off-center reflections in a mirrored circle; they can calso be regarded as perturbed rotations. Some of our experimental observations can be justified rigorously; for example, a lower bound is given for the number of cusps and the mode-locking behavior are studied. Symplectic topology is a particularly useful tool in this study.
We report on a compuational group theory experiment involving a search for cyclic presentations of the trivial group. The list of such presentations obtained includes counterexamples to a conjecture of M. J. Dunwoody.
We study the construction of auxiliary functions likely to aid in obtaining improved irrationality measures for cubic irrationalities and thence for arbitrary algebraic numbers. Specifically, we note that the construction of curves with singularities appropriately prescribed for our purpose leads to a simultaneous Padé approximation problem. The first step towards an explicit construction appears to be the evaluation of certain determinants. Our main task here is the computation of an example determinant, which turns out indeed to be a product of a small number of factors each to high multiplicity--whence the adjective 'powerful'. Our evaluation confirms a computational conjecture of Bombieri, Hunt and van der Poorten.