Pure product polynomials and the Prouhet-Tarry-Escott problem

An -factor pure product is a polynomial which can be expressed in the form for some natural numbers . We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every -factor pure product has norm at least . We describe three algorithms for determining the least norm an -factor pure product can have. We report results of our computations using one of these algorithms which include the result that every -factor pure product has norm strictly greater than if is , , , or .

     

Numerical solution of the scalar double-well problem allowing microstructure

The direct numerical solution of a non-convex variational problem () typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem () leading to a (degenerate) convex minimisation problem. The problem () has a minimiser and a related stress field which is known to coincide with the stress field obtained by solving () in a generalised sense involving Young measures. If is a finite element solution, is the related discrete stress field. We prove a priori and a posteriori estimates for in and weaker weighted estimates for . The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

The exponent of discrepancy is at most

We study discrepancy with arbitrary weights in the norm over the -dimensional unit cube. The exponent of discrepancy is defined as the smallest for which there exists a positive number such that for all and all there exist points with discrepancy at most . It is well known that . We improve the upper bound by showing that

This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent is .

     

Modular forms which behave like theta series

In this paper, we determine all modular forms of weights , , for the full modular group which behave like theta series, i.e., which have in their Fourier expansions, the constant term and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in (resp. in ) for the cases (resp. for the cases ). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

     

Bounds for Multiplicative Cosets over Fields of Prime Order

Let be a positive integer and suppose that is an odd prime with . Suppose that and consider the polynomial . If this polynomial has any roots in , where the coset representatives for are taken to be all integers with , then these roots will form a coset of the multiplicative subgroup of consisting of the th roots of unity mod . Let be a coset of in , and define . In the paper ``Numbers Having Small th Roots mod ' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for of the form , where is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when is a power of or when

     

The practical computation of areas associated with binary quartic forms

We derive formulas for practically computing the area of the region defined by a binary quartic form . These formulas, which involve a particular hypergeometric function, are useful when estimating the number of lattice points in certain regions of the type and will likely find application in many contexts. We also show that for forms of arbitrary degree, the maximal size of the area of the region , normalized with respect to the discriminant of and taken with respect to the number of conjugate pairs of , increases as the number of conjugate pairs decreases; and we give explicit numerical values for these normalized maxima when is a quartic form.

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

On Wendt's determinant

Wendt's determinant of order is the circulant determinant whose -th entry is the binomial coefficient , for . We give a formula for , when is even not divisible by 6, in terms of the discriminant of a polynomial , with rational coefficients, associated to . In particular, when where is a prime , this yields a factorization of involving a Fermat quotient, a power of and the 6-th power of an integer.

     

Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turán concerning the `distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial of degree there exists a monic polynomial with deg() = deg() such that is irreducible over and the `distance' of and is .

     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.

     

On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the -th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of , the billionth hexadecimal digits of and , and the ten billionth decimal digit of . These calculations rest on the observation that very special types of identities exist for certain numbers like , , and . These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for :

     

Computing Canonical Heights With Little (or No) Factorization

Let be an elliptic curve with discriminant , and let . The standard method for computing the canonical height is as a sum of local heights . There are well-known series for computing the archimedean height , and the non-archimedean heights are easily computed as soon as all prime factors of have been determined. However, for curves with large coefficients it may be difficult or impossible to factor . In this note we give a method for computing the non-archimedean contribution to which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.

     

A search for Wieferich and Wilson primes

An odd prime is called a Wieferich prime if

alternatively, a Wilson prime if

To date, the only known Wieferich primes are and , while the only known Wilson primes are , and . We report that there exist no new Wieferich primes , and no new Wilson primes . It is elementary that both defining congruences above hold merely (mod ), and it is sometimes estimated on heuristic grounds that the ``probability" that is Wieferich (independently: that is Wilson) is about . We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod ).

     

On some inequalities for the incomplete gamma function

Let be a positive real number. We determine all real numbers and such that the inequalities

are valid for all . And, we determine all real numbers and such that

hold for all .

     

Approximating the number of integers free of large prime factors

Define to be the number of positive integers such that has no prime divisor larger than . We present a simple algorithm that approximates in floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the well-known approximation , where is Dickman's function.

     

On the -divisibility of Fermat quotients

The authors carried out a numerical search for Fermat quotients vanishing mod , for , up to . This article reports on the results and surveys the associated theoretical properties of . The approach of fixing the prime rather than the base leads to some aspects of the theory apparently not published before.

     

A sinc-collocation method for initial value problems

A collocation procedure is developed for the initial value problem , , using the globally defined sinc basis functions. It is shown that this sinc procedure converges to the solution at an exponential rate, i.e., where and basis functions are used in the expansion. Problems on the domains and are used to illustrate the implementation and accuracy of the procedure.

     

A continuity property of multivariate Lagrange interpolation

Let be a sequence of interpolation schemes in of degree (i.e. for each one has unique interpolation by a polynomial of total degree and total order . Suppose that the points of tend to as and the Lagrange-Hermite interpolants, , satisfy for all monomials with . Theorem: for all functions of class in a neighborhood of . (Here denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

     

Further tabulation of the Erdös-Selfridge function

For a positive integer , the Erdös-Selfridge function is the least integer such that all prime factors of exceed . This paper describes a rapid method of tabulating using VLSI based sieving hardware. We investigate the number of admissible residues for each modulus in the underlying sieving problem and relate this number to the size of . A table of values of for is provided.