The minimal number of solutions to

In 1958, A. Schinzel showed that for each fixed there are at least two solutions to . Using the same method and a computer search, Schinzel and A. Wakulicz extended the bound to all . Here we show that Schinzel's method can be used to further extend the bound when is even, but not when is odd.

     

Finding strong pseudoprimes to several bases. II

Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Upper bounds for were first given by Jaeschke, and those for were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

We conjecture that

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.

     

Coding the principal character formula for affine Kac-Moody lie algebras

In this paper, an algorithm for computing the principal character for affine Lie algebras is discussed and presented. The principal characters discovered using this program are given and/or proven. Results include level 2 and 3 character formulas in and the sole existence of the Rogers-Ramanujan products in , , , , , , .     

Analysis of a bilinear finite element for shallow shells. II: Consistency error

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound can be obtained if the contribution of transverse shear is neglected. Here is the thickness of the shell, the mesh spacing, and a smoothness parameter. In the bending-dominated case, the uniformly optimal bound is achievable but requires that membrane and transverse shear strains are of order as . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound .

     

Computation of Stark-Tamagawa units

Let be a totally real number field and let denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the ``Equivariant Tamagawa Number Conjecture' for the -equivariant motive , where is a cyclic extension of degree and group . This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet -functions.

     

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter . The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by with independent of the mesh parameter , the diffusion coefficient and the exact solution of the problem.

     

An estimate for the number of integers without large prime factors

denotes the number of positive integers and free of prime factors y$">. Hildebrand and Tenenbaum provided a good approximation of . However, their method requires the solution to the equation , and therefore it needs a large amount of time for the numerical solution of the above equation for large . Hildebrand also showed approximates for , where and is the unique solution to . Let be defined by for 0$">. We show approximates , and also approximates , where . Using these approximations, we give a simple method which approximates within a factor in the range , where is any positive constant.

     

Error bounds for Gauss-Turán quadrature formulae of analytic functions

We study the kernels of the remainder term of Gauss-Turán quadrature formulas

for classes of analytic functions on elliptical contours with foci at , when the weight is one of the special Jacobi weights ; ; , ; , . We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.

     

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.

     

Lower bounds for the total stopping time of iterates

The total stopping time of a positive integer is the minimal number of iterates of the function needed to reach the value , and is if no iterate of reaches . It is shown that there are infinitely many positive integers having a finite total stopping time such that 6.14316 \log n.$"> The proof involves a search of trees to depth 60, A heuristic argument suggests that for any constant , a search of all trees to sufficient depth could produce a proof that there are infinitely many such that \gamma\log n.$">It would require a very large computation to search trees to a sufficient depth to produce a proof that the expected behavior of a ``random' iterate, which is occurs infinitely often.

     

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 .

     

A computational approach for solving

We present a computational approach for finding all integral solutions of the equation for even values of . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for assuming the Generalized Riemann Hypothesis, and for unconditionally.

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests indicate that the rate is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the -norm. Furthermore, under the same regularity assumption, an -uniform convergence of order in the norm is proved for some mesh points in the boundary layer region.

     

Biquadratic reciprocity and a Lucasian primality test

Let be the sequence defined from a given initial value, the seed, , by the recurrence . Then, for a suitable seed , the number (where is odd) is prime iff . In general depends both on and on . We describe a slight modification of this test which determines primality of numbers with a seed which depends only on , provided . In particular, when , odd, we have a test with a single seed depending only on , in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for , Math. Comp. 61 (1993), 97-109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity.

     

New quadratic polynomials with high densities of prime values

Hardy and Littlewood's Conjecture F implies that the asymptotic density of prime values of the polynomials , is related to the discriminant of via a quantity The larger is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant . A technique of Bach allows one to estimate accurately for any , given the class number of the imaginary quadratic order with discriminant , and for any 0$"> given the class number and regulator of the real quadratic order with discriminant . The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants for which is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of efficiently, even for values of with as many as decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis, is larger than any previously known examples.

     

Real zeros of real odd Dirichlet -functions

Let be a real odd Dirichlet character of modulus , and let be the associated Dirichlet -function. As a consequence of the work of Low and Purdy, it is known that if and , , , then has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if , then has no positive real zeros.

     

More on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This paper extends these results to show that .