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 .

     

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 .

     

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.

     

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.

     

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.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

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 .

     

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 .

     

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

Random Fibonacci sequences and the number

For the familiar Fibonacci sequence (defined by , and for ), increases exponentially with at a rate given by the golden ratio . But for a simple modification with both additions and subtractions - the random Fibonacci sequences defined by , and for , , where each sign is independent and either or - with probability - it is not even obvious if should increase with . Our main result is that

with probability . Finding the number involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.

     

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.

     

Zeros of 2-adic -functions and congruences for class numbers and fundamental units

We study the imaginary quadratic fields such that the Iwasawa -invariant equals 1, obtaining information on zeros of -adic -functions and relating this to congruences for fundamental units and class numbers.

     

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 .

     

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.

     

Calculation of values of -functions associated to elliptic curves

We calculated numerically the values of -functions of four typical elliptic curves in the critical strip in the range . We found that all the non-trivial zeros in this range lie on the critical line and are simple except the one at . The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane , which enables us to calculate for large . Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

     

A new bound for the smallest

Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.

     

On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

We consider the convergence of Gauss-type quadrature formulas for the integral , where is a weight function on the half line . The -point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials }, where is a sequence of integers satisfying and . It is proved that under certain Carleman-type conditions for the weight and when or goes to , then convergence holds for all functions for which is integrable on . Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.

     

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an -regular solution, a first-order error bound in the norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an -regular solution, first-order convergence in the norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and -conditional stability for error propagation, where for the Schrödinger-Poisson system and for the cubic Schrödinger equation.