Showing papers on "Prime number published in 1998"
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11 Jun 1998
TL;DR: The book "Metric Number Theory" covers a wide range of topics in number theory, including normal numbers, Diophantine approximation, uniform distribution, and exceptional sets of measure zero.
Abstract: Abstract This book deals with the number-theoretic properties of almost all real numbers. It brings together many different types of result never covered within the same volume before, thus showing interactions and common ideas between different branches of the subject. It provides an indispensable compendium of basic results, important theorems and open problems. Starting from the classical results of Borel, Khintchine and Weyl, normal numbers, Diophantine approximation and uniform distribution are all discussed. Questions are generalized to higher dimensions and various non-periodic problems are also considered (for example restricting approximation to fractions with prime numerator and denominator). Finally, the dimensions of some of the exceptional sets of measure zero are considered.
72 citations
TL;DR: In this paper, it was shown that the problem of finding the least k such that the inequality with fixed e > 0 has solutions in prime numbers for every sufficiently large real N is solvable if k ≥ k 0 and N is sufficiently large.
Abstract: where c > 1 is not integer, and proved in both cases that there exists k0(c) such that the corresponding problem has solutions if k ≥ k0 and N is sufficiently large. Later Deshouillers [4] and Arhipov and Zhitkov [1] improved Segal’s result on (2). One may also mention the papers of Deshouillers [5] and Gritsenko [7], where the equation (2) in two variables was considered. In 1952 I. I. Piatetski–Shapiro [12] considered (1) with x1, . . . , xk restricted to prime numbers. Let H(c) denote the least k such that the inequality (1) with fixed e > 0 has solutions in prime numbers for every sufficiently large real N . Piatetski–Shapiro proved that
53 citations
TL;DR: In this paper, it was proved that for any integer k ≥ 54 000, there is n ≥ n ≥ k ≥ 0 depending onk only such that every even integer ≥ n k ≥ k is a sum of two odd prime numbers andk powers of 2.
Abstract: It is proved that for any integerk≥ 54 000, there isN
k
>0 depending onk only such that every even integer ≥N
k
is a sum of two odd prime numbers andk powers of 2.
45 citations
40 citations
TL;DR: In this paper, it was shown that any system of homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (r2)2d-1.
Abstract: Let p be a rational prime number. We refine Brauer's elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd2)2d-1. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.
39 citations
Patent•
1 Dec 1998TL;DR: In this paper, a pseudo-random sequence of integers is generated by selecting the least significant bit from each integer value and then used to encrypt a message using a selected encryption algorithm such as the XOR algorithm.
Abstract: A method is provided for generating a pseudo-random sequence of integers, and the method is applied to the encryption of messages. The method uses a key K and a pair of prime numbers p and q, where q=2p+1. According to one aspect of the invention, a sequence of integers is formed. A sequence of bits is then formed from the sequence of integers, e.g., by selecting the least significant bit from each integer value. The sequence of bits is then used to encrypt a message using a selected encryption algorithm such as the XOR algorithm. Since prime numbers p and q can be selected to be larger than key K, the repeating period of the sequence of integers is larger than that permitted by the bit length of K.
38 citations
Patent•
IBM1
TL;DR: An access control mechanism using a grouping system whereby each group is assigned a unique prime number is described in this article, where the resource objects to be accessed are assigned a value that is determined by multiplying all of the group prime numbers from the groups that have access to that resource.
Abstract: An access control mechanism using a grouping system whereby each group is assigned a unique prime number. The resource objects to be accessed are assigned a value that is determined by multiplying all of the group prime numbers from the groups that have access to that resource. Also, each user is assigned to one or more groups and each user has an access number that is a product of the prime numbers assigned to each group. When a particular user desires access to a particular resource object, the greatest common divisor between the resource product and the user product is determined. If the resulting greatest common divisor is greater than one, then the user is allowed access. If the greatest common divisor is one (the lowest prime), the user is denied access.
37 citations
21 Jun 1998
TL;DR: In this paper, the Goldbach conjecture was verified for all the even numbers in the intervals [105i, 105i + 108], for i=3, 4,..., 20 and [1010i, 1010i + 109] for i =20,21,..., 30.
Abstract: The Goldbach conjecture states that every even integer ≥ 4 can be written as a sum of two prime numbers. It is known to be true up to 4 × 1011. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R10000 CPUs are described, which extend this bound to 1014. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number ≥7 can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [105i , 105i +108], for i=3, 4,..., 20 and [1010i , 1010i + 109], for i=20,21,..., 30.
30 citations
Patent•
IBM1
TL;DR: A method, system and apparatus for generating primes (p and q) for use in cryptography from secret random numbers and an initialization value whereby the initial secret random number are encoded into the generated primes is described in this paper.
Abstract: A method, system and apparatus for generating primes (p and q) for use in cryptography from secret random numbers and an initialization value whereby the initial secret random numbers are encoded into the generated primes. This eliminates the need to retain the initial secret random numbers for auditing purposes. The initialization value may also be generated from information readily available, if so desired, resulting in additional entropy without the requirement of storing additional information.
30 citations
TL;DR: A trivial necessary condition and a sufficient condition are given for the factorization of a bipartite multigraph λ K m,n into copies of K 1, p 2 whenever p is a prime number.
20 citations
TL;DR: A search for prime factors of the generalized Fermat numbers F n (a,b) = a 2n + b 2n has been carried out and all numbers with n ≤ 7 are now completely factored.
Abstract: A search for prime factors of the generalized Fermat numbers F n (a,b) = a 2n + b 2n has been carried out for all pairs (a, b) with a, b < 12 and GCD(a, b) = 1. The search limit k on the factors, which all have the form p = k 2 m + 1, was k = 10 9 for m < 100 and k = 3. 10 6 for 101 < m < 1000. Many larger primes of this form have also been tried as factors of F n (a, b). Several thousand new factors were found, which are given in our tables. For the smaller of the numbers, i.e. for n < 15, or, if a. b < 8, for n < 16, the cofactors, after removal of the factors found, were subjected to primality tests, an if composite with n ≤ 11, searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with n ≤ 7 are now completely factored.
Patent•
8 Apr 1998TL;DR: In this paper, a method of compressing data utilizes the prime number series to generate unique compression parameters that may be used to recover an original data stream, which is converted from a binary form to a decimal form.
Abstract: A method of compressing data utilizes the prime number series to generate unique compression parameters that may be used to recover an original data stream. The original data is converted from a binary form to a decimal form. Various compression parameters are selected to initialize the system. The compression parameters include the number of prime numbers which will be used in the compression process and an exponential value corresponding to each prime number. A header is constructed which includes the compression parameters. The data is compressed by a compression algorithm which performs successive division operations by the series of prime numbers selected. The compression algorithm generates a plurality of exponential values corresponding to each of the prime numbers. The header is then assembled with the exponential values and transmitted to a receiving station. In decompressing the data, the prime numbers are raised to the exponential value generated by the compression algorithm, and their product is taken. A fault parameter may be generated to compensate for numbers not readily divisible by the prime number selected in the initialization parameters.
TL;DR: In this article, the p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's padic class number formula, and express special values of padic L-functions in terms of orders of K-groups and higher padic regulators.
Abstract: In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with etale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters.
18 Oct 1998
TL;DR: This paper itemize three classes of security failures and formulate a simple algorithm for factoring n with a disclosed non-trivial factor of o(n) where the disclosure is for making use of a prime order subgroup in Z n * .
Abstract: Many cryptographic protocols and cryptosystems have been proposed to make use of prime order subgroups of Zn* where n is the product of two large distinct primes. In this paper we analyze a number of such schemes. While these schemes were proposed to utilize the difficulty of factoring large integers or that of finding related hidden information (e.g., the order of the group Zn*), our analyzes reveal much easier problems as their real security bases. We itemize three classes of security failures and formulate a simple algorithm for factoring n with a disclosed non-trivial factor of Φ(n) where the disclosure is for making use of a prime order subgroup in Zn* . The time complexity of our algorithm is O(n1/4/f) where f is a disclosed subgroup order. To factor such n of length up to 800 bits with the subgroup having a secure size against computing discrete logarithm, the new algorithm will have a feasible running time on use of a trivial size of storage.
TL;DR: In this article, the authors present conditions générales d'utilisation of commercial or impression systématique, constitutive of an infraction pénale.
Abstract: © Université Bordeaux 1, 1998, 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/conditions). 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.
TL;DR: The prime-number sequence, viewed as the spectrum of eigenvalues of random matrices, is found to be quasi-chaotic in this article, which supports the conjecture that a formula for the n-th prime does not exist.
Abstract: The prime-number sequence, viewed as thespectrum of eigenvalues of random matrices, is found tobe quasi-chaotic. Plots of histograms of prime-numbernearest-neighbor spacing Delta p at various values of total number of integers indicate roughagreement with the Wigner distribution and illustratelevel repulsion. A global maximum of these curves isnoted at Δp = 6. Numerical work further implies that in any maximum integer sampling, no matterhow large, a finite number of nearest neighbor spacingsdo not occur. This quasichaotic property of theprime-number sequence supports the conjecture that a formula for the n-th prime does not exist. Arule for missing spacings is inferred according towhich, as maximum number of integers \(\mathcal{N}\)increases, nearest neighbor vacancies corresponding tosmaller \(\mathcal{N}\) vanish and new, larger value vacancies appear. Inaddition, early values of these histograms illustrate arough oscillatory behavior with periodicityδ[Δp] ≃ 6. A corollary to the resultsimplies that zeros of the Riemann zeta function likewise comprisea quasi-chaotic sequence. Application of these findingsto the resonant spectra of excited nuclei isnoted.
21 Jun 1998
TL;DR: A randomized algorithm that takes as input a prime number p, and an algebraic set (represented by a system of polynomials) over the finite field Fp, and counts approximately the number of Fp-rational points in the set, which results in an asymptotic lower bound counting result.
Abstract: We present a randomized algorithm that takes as input a prime number p, and an algebraic set (represented by a system of polynomials) over the finite field Fp, and counts approximately the number of Fp-rational points in the set. For a fixed number of variables, the algorithm runs in random polynomial time with parallel complexity polylogarithmic in the input parameters (number of input polynomials, their maximum degree, and the prime p), using a polynomial number of processors. However, the degree of the polynomial bound on the running time grows sharply with the number of variables. A combinatorial analysis of the algorithm also shows that, when p is sufficiently large, a good approximate count is represented by Np D , where D is the highest possible dimension of an Fp-irreducible subvariety of the input defined over Fp, and N is the number of such distinct subvarieties. In addition, the algorithm computes these two numbers efficiently. It is also applied to obtain an asymptotic lower bound counting result in the case when an algebraic set defined over ℚ is reduced mod p, where p goes to infinity.
TL;DR: Criteria of divisibility of the class number h + of the real cyclotomic field Q(ζ p + ζ p -1 ) of a prime conductor p and of aprime degree l by primes q the order modulo I of which is l-1/2, are given.
Abstract: In this paper, criteria of divisibility of the class number h + of the real cyclotomic field Q(ζ p + ζ p -1 ) of a prime conductor p and of a prime degree l by primes q the order modulo I of which is l-1/2, are given A corollary of these criteria is the possibility to make a computational proof that a given q does not divide h + for any p (conductor) such that both p-1/2, p-3/4 are primes Note that on the basis of Schinzel's hypothesis there are infinitely many such primes p
1 Jan 1998
TL;DR: In this paper, a connected reductive algebraic group over k is defined as an algebraic closure of a finite field F q with q elements, where q is a fixed prime number, invertible in F q.
Abstract: Let G be a connected reductive algebraic group over k, an algebraic closure of a finite field F q with q elements. Assume that we are given an F q -rational structure on G with Frobenius map F: G → G. Let \(\bar{\mathbb{Q}}{{}_{l}}\) be an algebraic closure of the l-adic numbers (l is a fixed prime number, invertible in F q ).
TL;DR: In this article, it was shown that a positive proportion of the Hecke polynomials T 2k p (x) are irreducible if there are two distinct primes l and q.
21 Jun 1998
TL;DR: The concept of regular and irregular primes is extended to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as “higher Bernoulli numbers”.
Abstract: The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.
TL;DR: An analogue for composite moduli of the Wilson quotient is investigated and eight new Wilson numbers are found.
Abstract: An analogue for composite moduli of the Wilson quotient is
investigated. The Wilson numbers are studied and eight new
Wilson numbers are found.
TL;DR: A criterion of Inkeri is improved and it is shown that if there is a solution to Catalan's equation (1) x p - y q = ±1, with p andq prime numbers greater than 3 and both congruent to 3 (mod 4), then p and q form a double Wieferich pair.
Abstract: We improve a criterion of Inkeri and show that if there is a solution to Catalan's equation (1) x p - y q = ±1, with p and q prime numbers greater than 3 and both congruent to 3 (mod 4), then p and q form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 (mod 4). Indeed, in light of the results proved here it is reasonable to suppose that if q ≡ 3 (mod 4), then p and q form a double Wieferich pair.
1 Jan 1998
TL;DR: For a fixed real abelian field k, this article showed that the conjecture is "affirmative" for infinitely many p (which split in k) if we assume the abc conjecture for k.
Abstract: For any totally real number field k and any prime number p, Greenberg's conjecture for (k,p) asserts that the Iwasawa invariants Ap(k) and tip(k) are both zero. For a fixed real abelian field k, we prove that the conjecture is "affirmative" for infinitely many p (which split in k) if we assume the abc conjecture for k.
TL;DR: In this paper, the authors define a prime decomposition for a θ n -curve by 2-spheres, each of which intersects each edge of the curve at exactly one point.
TL;DR: The condition for isomorphism between double loops (or chordal rings) which is a simple kind of circulant graph is studied to show that the family of n nodes double loops having chord length below min{ n 4 , φ(n) 2 } has the Adam property.
TL;DR: In this paper, it was shown that there are infinitely many irreducible monics P in R with p ǫ h + T (P ) under some assumptions on q.
TL;DR: Inferior Smarandache Prime Part: For any positive real number n, one defines ISp(n) as the largest prime number less than or equal to n as discussed by the authors.
Abstract: Inferior Smarandache Prime Part: For any positive real number n one defines ISp(n) as the largest prime number less than or equal to n.
Book•
1 Jan 1998
TL;DR: Invited Talk 1: Shimura Curve Computations Noam D. Elkies (Harvard University) Invited Talk 2: The Decision Diffie-Hellman Problem Dan Boneh (Stanford University) GCD Algorithms Parallel Implementation of Schonhages Integer GCDAlgorithm Giovanni Cesari (Universita degli Studi di Trieste)
Abstract: Invited Talk 1: Shimura Curve Computations Noam D. Elkies (Harvard University) Invited Talk 2: The Decision Diffie-Hellman Problem Dan Boneh (Stanford University) GCD Algorithms Parallel Implementation of Schonhages Integer GCD Algorithm Giovanni Cesari (Universita degli Studi di Trieste) The Complete Analysis of the Binary Euclidean Algorithm Brigitte Vallee (Universite de Caen) Primality Cyclotomy Primality Proving -- Recent Developments Preda Mihailescu (FingerPIN AG & ETH, Institut fur wissentschaftliches Rechnen) Primality Proving Using Elliptic Curves: An Update F. Morain (Laboratoire d informatique de l'Ecole Polytechnique) Factoring Bounding Smooth Integers (extended abstract) Daniel J. Bernstein (The University of Illinois at Chicago) Factorization of the Numbers of the Form m 3 + c 2 m 2 + c 1 m+ c 0 Zhang Mingzhi (Sichuan Union University) Modelling the Yield of Number Field Sieve Polynomials Brian Murphy (Australian National University) A Montgomery-Like Square Root for the Number Field Sieve Phong Nguyen (Ecole Normale Superieure) Sieving Robert Bennions Hopping Sieve William F. Galway (University of Illinois at Urbana-Champaign) Trading Time for Space in Prime Number Sieves Jonathan P. Sorenson (Butler University) Analytic Number Theory Do Sums of 4 Biquadrates Have a Positive Density? Jean-Marc Deshouillers, Francois Hennecart, Bernard Landreau (Universite Bordeaux) New Experimental Results Concerning the Goldbach Conjecture J-M. Deshouillers (Universite Bordeaux), H.J.J. te Riele (CWI), Y. Saouter (Institut de Recherche en Informatique de Toulouse) Dense Admissible Sets Daniel M. Gordon, Gene Rodemich (Center for Communications Research) An Analytic Approach to Smooth Polynomials over Finite Fields Daniel Panario (University of Toronto), Xavier Gourdon (INRIA), Philippe Flajolet (INRIA) Cryptography Generating a Product of Three Primes with an Unknown Factorization Dan Boneh, Jeremy Horwitz (Stanford University) On the Performance of SignatureSchemes Based on Elliptic Curves Erik De Win (Katholieke Universiteit Leuven), Serge Mister (Queens University) Bart Preneel (Katholieke Universiteit Leuven), Michael Wiener (Entrust Technologies) NTRU: A Ring-Based Public Key Cryptosystem Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman (Brown University) Finding Length-3 Positive Cunningham Chains and their Cryptographic Significance Adam Young (Columbia University), Moti Yung (CertCo) Linear Algebra, Lattices Reducing Ideal Arithmetic to Linear Algebra Problems Stefan Neis (Darmstadt University of Technology) Evaluation of Linear Relations between Vectors of a Lattice in Euclidean Space I. A. Semaev An Efficient Parallel Block-Reduction Algorithm Susanne Wetzel (Universitat des Saarlandes) Series, Sums Fast Multiprecision Evaluation of Series of Rational Numbers Bruno Haible (ILOF), Thomas Papanikolaou (Laboratoire A2X) A Problem Concerning a Character Sum --- Extended Abstract E. Teske (Technische Universitat Darmstadt), H.C. Williams (University of Manitoab) Formal Power Series and Their Continued Fraction Expansion Alf van der Poorten (Centre for Number Theory Research) Algebraic Number Fields Imprimitive Octic Fields with Small Discriminants Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (Universite Bordeaux I) A Table of Totally Complex Number Fields of Small Discriminants Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (Universite Bordeaux I) Generating Arithmetically Equivalent Number Fields with Elliptic Curves Bart de Smit (Rijksuniversiteit Leiden) Computing the Lead Term of an Abelian L-Function David S. Dummit (University of Vermont), Brett A. Tangedal (College of Charleston Timing Analysis of Targeted Hunter Searches John W. Jones (Arizona State University), David P. Roberts (Rutgers University) On Successive Minima of Rings of Algebraic Integers Jacques Martinet (Universite Bordeaux I) Class Groups and Fields Computation of Relative Quadratic Class Groups Henri Cohen, Francisco Diaz y
1 Jan 1998