Showing papers on "Integer published in 1993"

Exponential sums and lattice points III

Abstract: The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

Integers without large prime factors

Abstract: © Université Bordeaux 1, 1993, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The structure of random partitions of large integers

Abstract: Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches ∞. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct

Generalization of a problem of Diophantus

Abstract: Greek mathematician Diophantus of Alexandria noted that the numbers 1 16 , 33 16 , 68 16 and 105 16 have the property that the product of any two of them when increased by 1 is a square of a rational number. Let n be an integer. We say the set of natural numbers {a1, a2, . . . , am} has the property of Diophantus of order n, in brief D(n), if for all i, j = 1, 2, . . . ,m, i 6= j, the following holds: ai · aj + n = bij , where bij is an integer. The first set of four natural numbers with property D(1) was found by French mathematician Pierre de Fermat (1601 1665). That set is {1, 3, 8, 120}. Davenport and Baker [3] show that a fifth integer r cannot be added to that set and maintain the same property unless r = 0. For the rational number r = 777480 8288641 the product of any two different members of that set increased by 1 is the square of a rational number (see [1]). In this paper we consider some problems of existence of sets of four natural numbers with property D(n), for any integer n. We prove that, for all e ∈ Z, there exist infinite numbers of sets of four natural numbers with property D(e2). Indeed, we show how a set {a, b} with property D(e2) can be extended to a set {a, b, c, d} with the same property, if a · b is not a perfect square. That construction is applied to the identities

Static and Dynamic Algorithms for k-Point Clustering Problems

Abstract: Let S be a set of n points in d-space, where d ≥ 2 is a constant, and let 1 ≤ k ≤ n be an integer. A unified approach is given for solving the problem of finding a subset of S of size k that minimizes some closeness measure, such as the diameter, perimeter or the circumradius. Moreover, data structures are given that maintain such a subset under insertions and deletions of points.

On strong pseudoprimes to several bases

Abstract: With Y'k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5, q6, q7, q8 and give upper bounds for V/9, / W t,' 1 . We discuss the methods and underlying facts for obtaining these results. 1. PRIMALITY TESTS BY MEANS OF STRONG PSEUDOPRIMES Computer algebra systems, as for instance AXIOM [2], use strong pseudoprimes for testing primality of integers. The advantage of such tests is that they are very efficient. The disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals. To make such tests deterministic for integers in prescribed intervals, one has to know the exact number of necessary so-called "strong pseudoprimality tests". For this purpose we introduce the numbers V1i, V/2, . .. for which we compute lower and upper bounds. These numbers are defined and discussed in this section; in ?2 we derive some facts which are the basis for finding bounds for the numbers V/k. In ?3 we discuss the methods which led to our results. In view of Fermat's "Little Theorem" we know that n is certainly not a prime when we have bn-1 i 1 mod n for an integer b with 1 0, and when n is a composite number, then n is called a "strong pseudoprime to base b" if either

Factoring with cubic integers

Abstract: We describe an experimental factoring method for numbers of form x3+k; at present we have used only k=2. The method is the cubic version of the idea given by Coppersmith, Odlyzko and Schroeppel (Algorithmica 1 (1986), 1–15), in their section ‘Gaussian integers’. We look for pairs of small coprime integers a and b such that: i. the integer a+bx is smooth, ii. the algebraic integer a+bz is smooth, where z3=−k. This is the same as asking that its norm, the integer a3 - kb3 shall be smooth (at least, it is when k=2).

A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem

Abstract: We describe a time randomized algorithm that estimates the number of feasible solutions of a multidimensional knapsack problem within 1 ± e of the exact number (Here r is the number of constraints and n is the number of integer variables) The algorithm uses a Markov chain to generate an almost uniform random solution to the problem

Lower bounds for arithmetic networks

Abstract: We show lower bounds for depth of arithmetic networks over algebraically closed fields, real closed fields and the field of the rationals. The parameters used are either the degree or the number of connected components. These lower bounds allow us to show the inefficiency of arithmetic networks to parallelize several natural problems. For instance, we show a √n lower bound for parallel time of the Knapsack problem over the reals and also that the computation of the “integer part” is not well parallelizable by arithmetic networks. Over the rationals we obtain results of similar order and that the Knapsack has an √n lower bound for the parallel time measured by networks. A simply exponential lower bound for the parallel time of quantifier elimination is also shown. Finally, separations among classesP K andNC K are available for fieldsK in the above cases.

Alkylmethylsiloxane containing perfume compositions

Abstract: A composition which is an emulsifier-free mixture of a perfume oil and a cyclic alkylmethylsiloxane having the formula (see formula I) in which x and y are each integers and the sum of x and y is four, five or six. with the proviso that x and y cannot be zero; and z is an integer having a value of 5-50.

On the distribution of k -dimensional vectors for simple and combined Tausworthe sequences

Abstract: The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F2. The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into 2kl identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.

Approximate and Exact Deterministic Parallel Selection

Abstract: The selection problem of size n is, given a set of n elements drawn from an ordered universe and an integer r with 1 0 asks for any element whose true rank differs from r by at most An. Our main results are: (1) For all t≥(log log n)4, approximate selection problems of size n can be solved in O(t) time with optimal speedup with relative accuracy \(2^{{{ - t} \mathord{\left/{\vphantom {{ - t} {\left( {\log \log n} \right)}}} \right.\kern- ulldelimiterspace} {\left( {\log \log n} \right)}}^4 }\); no deterministic PRAM algorithm for approximate selection with a running time below Ο(log n/log log n) was previously known. (2) Exact selection problems of size n can be solved in O(log n/log log n) time with O(n log log n/log n) processors. This running time is the best possible (using only a polynomial number of processors), and the number of processors is optimal for the given running time (optimal speedup); the best previous algorithm achieves optimal speedup with a running time of O(log n log*n/log log n).

Constructing Small Sets that are Uniform in Arithmetic Progressions

Abstract: For every integer N , we explicitly construct a subset of residues mod N of size(log N ) o (1) which is nearly uniformly distributed in every arithmetic progression modulo N .

On the nonperiodic cyclic equivalence classes of Reed-Solomon codes

Abstract: Picking up exactly one member from each of the nonperiodic cyclic equivalence classes of an (n, k+1) Reed-Solomon code E over GF(q) gives a code, E", which has bounded Hamming correlation values and the self-synchronizing property. The exact size of E" is shown to be (1/n) Sigma /sub d mod n/ mu (d)q/sup 1+k/d/, where mu (d) is the Mobius function, (x) is the integer part of x, and the summation is over all the divisors d of n=q-1. A construction for a subset V of E is given to prove that mod E" mod >or= mod V mod =(q/sup k+1/-q/sup k+1-N/)/(q-1) where N is the number of integers from 1 to k which are relatively prime to q-1. A necessary and sufficient condition for mod E" mod = mod V mod is proved and some special cases are presented with examples. For all possible values of q>2, a number B(q) is determined such that mod E" mod = mod V mod for 1 mod V mod for k>B(q). >

A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares

Abstract: Diophantus probably knew, and Lagrange [L] proved, that every positive integer can be written as a sum of four perfect squares. Jacobi [J] proved the stronger result that the number of ways in which a positive integer can be so written1 equals 8 times the sum of its divisors that are not multiples of 4. Here we give a short new proof that only uses high school algebra, and is completely from scratch. All infinite series and products that appear are to be taken in the entirely elementary sense of formal power series. The problem of representing integers as sums of squares has drawn the attention of many great mathematicians, and we encourage the reader to look up Grosswald's [G] erudite masterpiece on this subject. The crucial part of our proof is played by two simple identities, that we state as one Lemma.

The Frobenius problem and maximal lattice free bodies

Abstract: Let p = (p1,…,pn,) be a vector of positive integers whose greatest common divisor is unity. The Frobenius problem is to find the largest integer f* which cannot be written as a nonnegative integral combination of the pi. In this note we relate the Frobenius problem to the topic of maximal lattice free bodies and describe an algorithm for n = 3.

Abstract properties of $ s$-arithmetic groups and the congruence problem

Abstract: Suppose is a simple and simply connected algebraic group over an algebraic number field and is a finite set of valuations of containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the -arithmetic subgroup and the congruence property, i.e. the finiteness of the corresponding congruence kernel . In particular, it is shown that if the profinite completion satisfies condition (PG), (i.e., for any integer > 0 and any prime there exist and such that for all 0$ SRC=http://ej.iop.org/images/1468-4810/40/3/A01/tex_im_2173_img12.gif/>), then is finite. Examples are given demonstrating the possibility of effectively verifying (PG)' .

Small Fractional Parts of Additive Forms

Abstract: We show how the methods of Vaughan & Wooley, which have proved fruitful in dealing with Waring’s problem, may also be used to investigate the fractional parts of an additive form. Results are obtained which are near to best possible for forms with algebraic coefficients. New results are also obtained in the general case. Extensions are given to make several additive forms simultaneously small. The key ingredients in this work are: mean value theorems for exponential sums, the use of a small common factor for the integer variables, and the large sieve inequality.

Poisson-Lie group of pseudodifferential symbols and fractional KP-KdV hierarchies

Abstract: The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the ``logarithmic'' 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the Lie group of classical symbols of all real (or complex) degrees. It turns out that this group has a natural Poisson-Lie structure whose restriction to differential operators of an arbitrary integer order coincides with the second Adler-Gelfand-Dickey structure. Moreover, for any real (or complex) \alpha there exists a hierarchy of completely integrable equations on the degree \alpha pseudodifferential symbols, and this hierarchy for \alpha=1 coincides with the KP one, and for an integer \alpha=n>1$ and purely differential symbol gives the n-KdV-hierarchy.

Rational approximations to the dilogarithm

Abstract: The irrationality proof of the values of the dilogarithmic function L 2 (z) at rational points z = 1/k for every integer k ∈ (−∞, −5] ∪ [7, ∞) is given. To show this we develop the method of Pade-type approximations using Legendre-type polynomials, which also derives good irrationality measures of L 2 (1/k). Moreover, the linear independence over Q of the numbers 1, log(1 − 1/k), and L 2 (1/k) is also obtained for each integer k ∈ (−∞, −5] ∪ [7, ∞)

On Parallel Integer Merging

Abstract: The problem of merging two sorted arrays A = (a1, a2, ..., an1) and B = (b1, b2, ..., bn2) is considered. For input elements that are drawn from a domain of integers 1...s] we present an algorithm that runs in O(log log log s) time using n/log log log s CREW PRAM processors (optimal speed-up) and O(ns?) space, where n = n1 + n2. For input elements that are drawn from a domain of integers 1...n] we present a second algorithm that runs in O(?(n)) time (where ?(n) is the inverse of Ackermann?s function) using n/?(n) CREW PRAM processors and linear space. This second algorithm is non-uniform; however, it can be made uniform at a price of a certain loss of speed, or by using a CRCW PRAM.

New invariants and class number problem in real quadratic fields

Abstract: In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit eD of real quadratic field Q(√D) satisfies NeD = –1, and studied relationships among these new invariants and already known invariants. One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NeD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

The analytic rank of a family of elliptic curves

Abstract: We study the family of elliptic curves Em X3 + Y3 = m where m is a cubefree integer The elliptic curves Em with even analytic rank and those with odd analytic rank are proved to be equally distributed It is proved that the number of cubefree integers m ≤ X such that the analytic rank of Em is even and ≥ 2 is at least CX 2/3-e, where e is arbitrarily small and C is a positive constant, for X large enough Therefore, if we assume the Birch and Swinnerton-Dyer conjecture, the number of all cubefree integers m ≤ X such that the equation X3 + Y3 = m have at least two independent rational solutions is at least CX2/3-e