New Fibonacci and Lucas primes

Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers have been determined for and all probable prime Lucas numbers have been determined for . A rigorous proof of primality is given for and for numbers with , , , , , , , , the prime having 3020 digits. Primitive parts and of composite numbers and have also been tested for probable primality. Actual primality has been established for many of them, including 22 with more than 1000 digits. In a Supplement to the paper, factorizations of numbers and are given for as far as they have been completed, adding information to existing factor tables covering .

     

Nonexistence conditions of a solution for the congruence

We obtain nonexistence conditions of a solution for of the congruence , where , and are integers, and is a prime power. We give nonexistence conditions of the form for , , , , , and of the form for , , , . Furthermore, we complete some tables concerned with Waring's problem in -adic fields that were computed by Hardy and Littlewood.

     

Computing automorphisms of abelian number fields

Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .

     

The number of primes is finite

For a positive integer let and let . The number of primes of the form is finite, because if , then is divisible by . The heuristic argument is given by which there exists a prime such that for all large ; a computer check however shows that this prime has to be greater than . The conjecture that the numbers are squarefree is not true because .

     

Solving polynomials by radicals with roots of unity in minimum depth

Let be an algebraic number field. Let be a root of a polynomial which is solvable by radicals. Let be the splitting field of over . Let be a natural number divisible by the discriminant of the maximal abelian subextension of , as well as the exponent of , the Galois group of over . We show that an optimal nested radical with roots of unity for can be effectively constructed from the derived series of the solvable Galois group of over .

     

Normal bases via general Gauss periods

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive th root of unity, one obtains under certain conditions a normal basis for over , where is a prime and for some integer . We generalize this construction by allowing arbitrary integers with , and find in many cases smaller values of than is possible with the previously known approach.

     

Analysis of PSLQ, an integer relation finding algorithm

Let be either the real, complex, or quaternion number system and let be the corresponding integers. Let be a vector in . The vector has an integer relation if there exists a vector , , such that . In this paper we define the parameterized integer relation construction algorithm PSLQ, where the parameter can be freely chosen in a certain interval. Beginning with an arbitrary vector , iterations of PSLQ will produce lower bounds on the norm of any possible relation for . Thus PSLQ can be used to prove that there are no relations for of norm less than a given size. Let be the smallest norm of any relation for . For the real and complex case and each fixed parameter in a certain interval, we prove that PSLQ constructs a relation in less than iterations.

     

Largest known twin primes and Sophie Germain primes

The numbers are twin primes. The number is a Sophie Germain prime, i.e. and are both primes. For , the numbers , and are all primes.

     

Double roots of power series and related matters

For a given collection of distinct arguments , multiplicities and a real interval containing zero, we are interested in determining the smallest for which there is a power series with coefficients in , and roots of order respectively. We denote this by . We describe the usual form of the extremal series (we give a sufficient condition which is also necessary when the extremal series possesses at least non-dependent coefficients strictly inside , where is 1 or 2 as is real or complex). We focus particularly on , the size of the smallest double root of a power series lying on a given ray (of interest in connection with the complex analogue of work of Boris Solomyak on the distribution of the random series ). We computed the value of for the rationals in of denominator less than fifty. The smallest value we encountered was . For the one-sided intervals and the corresponding smallest values were and .

     

The second largest prime divisor of an odd perfect number exceeds ten thousand

Let denote the sum of positive divisors of the natural number . Such a number is said to be perfect if . It is well known that a number is even and perfect if and only if it has the form where is prime.

No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number.

For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed , and Hagis showed that the second largest must exceed . In this paper, we improve the latter bound. In particular, we prove the statement in the title of this paper.

     

Zeta functions of a class of elliptic curves over a rational function field of characteristic two

We show how to calculate the zeta functions and the orders of Tate-Shafarevich groups of the elliptic curves with equation over the rational function field , where is a power of 2. In the range , , odd of degree , the largest values obtained for are (one case), (one case) and (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL over local or global fields of characteristic two.

     

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).

     

Elliptic binomial diophantine equations

The complete sets of solutions of the equation are determined for the cases , , , , , , , . In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case () we had to devise a new variant of the method.

     

On the discrete logarithm in the divisor class group of curves

Let be a curve which is defined over a finite field of characteristic . We show that one can evaluate the discrete logarithm in by operations in . This generalizes a result of Semaev for elliptic curves to curves of arbitrary genus.

     

Tables of linear congruential generators of different sizes and good lattice structure

We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For , we take as a modulus the largest prime smaller than , and provide a list of multipliers such that the MLCG with modulus and multiplier has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli , for multiplicative and non-multiplicative LCGs.

     

Computing rational points on rank 1 elliptic curves via -series and canonical heights

Let be an elliptic curve of rank 1. We describe an algorithm which uses the value of and the theory of canonical heghts to efficiently search for points in and . For rank 1 elliptic curves of moderately large conductor (say on the order of to ) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set contains non-torsion points.

     

Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, and . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for and . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: , and can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar - and -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

     

Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

We consider an abstract time-dependent, linear parabolic problem

where , , is a family of sectorial operators in a Banach space with time-independent domain . This problem is discretized in time by means of an A() strongly stable Runge-Kutta method, . We prove that the resulting discretization is stable, under the assumption

where and . Our results are applicable to the analysis of parabolic problems in the , , norms.

     

Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities

We consider the Poisson equation with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When , the rate of convergence to the singular solution in the energy norm is shown to be , and the rate of convergence to the stress intensity factors is shown to be , where is the largest re-entrant angle of the domain and can be arbitrarily small. The cost of the algorithm is . When , the algorithm can be modified so that the convergence rate to the stress intensity factors is . In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be .

     

A priori error estimates for Galerkin approximations to porous medium and fast diffusion equations

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

on bounded convex domains are considered. The range of the parameter includes the fast diffusion case . Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in norm with an error controlled by for and for . For the fully discrete problem, a global convergence rate of in norm is shown for the range . For , a rate of is shown in norm.