Showing papers on "Prime number published in 2000"
Book•
1 Feb 2000
TL;DR: In this paper, the authors present an extended version of the Euclidean Algorithm with the Lucas-Lehmer algorithm, which they call the Extended GCD Algorithm (EGCA).
Abstract: Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.
112 citations
Book•
5 May 2000TL;DR: The Riemann zeta function of prime numbers was studied in this paper, where it was shown that it is a stochastic distribution of the prime numbers, and the prime number theorem is proven.
Abstract: Genesis: From Euclid to Chebyshev The Riemann zeta function Stochastic distribution of prime numbers An elementary proof of the prime number theorem The major conjectures Further reading.
94 citations
17 Aug 2000
TL;DR: In this article, a simple way to substantially reduce the value of hidden constants is proposed to provide much more efficient prime number generation algorithms, which are applied to various contexts (DSA, safe primes, ANSI X9.31 compliant primes and strong primes).
Abstract: The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most real-life implementations are of rather poor performance. Common generators typically output a n-bit prime in heuristic average complexity O(n4) or O(n4/ log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms. We apply our techniques to various contexts (DSA primes, safe primes, ANSI X9.31-compliant primes, strong primes, etc.) and show how to build fast implementations on appropriately equipped smart-cards, thus allowing on-board key generation.
79 citations
Posted Content•
TL;DR: In this paper, it was shown that all indecomposable non-egenerate set-theoretical solutions to the quantum Yang-Baxter equation on a set of prime order are affine.
Abstract: In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such solutions. This result is a natural application of the general theory of set-theoretical solutions to the quantum Yang-Baxter equation. It is also a generalization of the corresponding statement for involutive set-theoretical solutions proved in an earlier paper of P.E. and A.S. with T.Schedler.
In order to prove our main result, we use the classification theory developed by the third author (based on the ideas of Lu, Yan, and Zhu) to reduce the problem to a group-theoretical statement: a finite group with trivial center generated by a conjugacy class of prime order is a subgroup of the affine group. Unfortunalely, we did not find an elementary proof of this statement, and our proof relies on the classification of outer automorphisms of finite simple groups.
78 citations
Book•
1 Jan 2000
TL;DR: The Fibonacci Numbers and the Arctic Ocean are represented by real numbers by means of fibonacci numbers as discussed by the authors, which is a representation of real numbers through means of the real numbers represented by the real number.
Abstract: The Fibonacci Numbers and the Arctic Ocean.- Representation of Real Numbers by Means of Fibonacci Numbers.- Prime Number Records.- Selling Primes.- Euler's Famous Prime Generating Polynomial and the Class Number of Imaginary Quadratic Fields.- Gauss and the Class Number Problem.- Consecutive Powers.- 1093.- Powerless Facing Powers.- What Kind of Number Is $$ \sqrt 2 ^{\sqrt 2 } $$ ?.- Galimatias Arithmeticae.
45 citations
Journal Article•
TL;DR: A simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms and how to build fast implementations on appropriately equipped smart-cards, thus allowing on-board key generation.
Abstract: The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most real-life implementations are of rather poor performance Common generators typically output a n-bit prime in heuristic average complexity O(n 4 ) or O(n 4 /log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms We apply our techniques to various contexts (DSA primes safe primes, ANSI X931-compliant primes, strong primes, etc) and show how to build fast implementations on appropriately equipped smart-cards, thus allowing on-board key generation
43 citations
TL;DR: An algorithmic procedure computing the polynomials and constants occurring in a Bezout identity, whose complexity is polynomial in the geometric degree and linear in the height of the system, is shown.
41 citations
1 Jan 2000
TL;DR: In this article, the authors studied Weil's well-known Explicit Formula in the theory of prime numbers and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis is true.
Abstract: — This Memoir studies Weil’s well-known Explicit Formula in the theory of prime numbers and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis is true. We prove that this quadratic functional attains its minimum in the unit ball of the L-space of functions with support in a given interval [−t; t ], and prove again Yoshida’s theorem that it is positive definite if t is sufficiently small. The Fourier transform of the functional gives rise to a quadratic form in infinitely many variables and we then study its finite truncations and corresponding eigenvalues. In particular, if the Riemann Hypothesis is false but only with finitely many non-trivial zeros off the critical line we show that the number of negative eigenvalues is precisely one-half of the number of zeros failing to satisfy the Riemann Hypothesis, provided the truncation is big enough.
39 citations
TL;DR: In this article, the authors generalize the Jouanolou non-integrability theorem concerning the system of ordinary differential equations to an arbitrary prime number n ≥ 3 of variables and arbitrary integer exponent s ≥ 3.
38 citations
TL;DR: For an algebraic number field k and a prime number p (if p=2, we assume that μ4⊂k), the maximal rank of a free pro-p extension of k was studied in this paper.
Abstract: For an algebraic number field k and a prime number p (if p=2, we assume that μ4⊂k), we study the maximal rank ρ
p
of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ
k
=r
2(k)+1 if and only if k is a so-called p-rational field.
33 citations
10 Dec 2000
TL;DR: A very efficient algorithm which given a negative integer Δ, Δ = 1 mod 8, Δ not divisible by 3, finds a prime number p and a cryptographically strong elliptic curve E over the prime field Fp whose endomorphism ring is the quadratic order O of discriminant Δ.
Abstract: We present a very efficient algorithm which given a negative integer Δ, Δ = 1 mod 8, Δ not divisible by 3, finds a prime number p and a cryptographically strong elliptic curve E over the prime field Fp whose endomorphism ring is the quadratic order O of discriminant Δ. Our algorithm bases on a variant of the complex multiplication method using Weber functions. We depict our very efficient method to find suitable primes for this method. Furthermore, we show that our algorithm is feasible in reasonable time even for orders O whose class number is in the range 200 up to 1000.
TL;DR: In this article, the authors proved a conjecture of Coleman, Kaskel, and Ribet about the ℚ-valued points of the modular curve X ∼ 0(N) which map to torsion points on J ∼ 0 (N) via the cuspidal embedding.
Abstract: Let N≥23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the ℚ-valued points of the modular curve X
0(N) which map to torsion points on J
0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X
0(N) into J
0(N).
TL;DR: In this paper, it was shown that f + pg is reducible in K [X ] for at most a finite number of primes p ∈ P ∈ X.
Posted Content•
[...]
TL;DR: In this paper, the p-adic valuation of the eigenvalue of Tp is defined as the slope of a modular form of weight k, level N, and trivial character which is an eigenform for the pth Hecke operator Tp.
Abstract: Fix a prime number p and choose, once and for all, an embedding of the algebraic closure of Q into Qp. Let k and N be integers, and suppose N is not divisible by p. If f is a modular form of weight k, level N, and trivial character which is an eigenform for the p-th Hecke operator Tp, we define the slope of f to be the p-adic valuation of the eigenvalue of Tp. This paper reports on computations that suggest that there is quite a lot of structure to the set of slopes for eigenforms of varying weight k. In particular, we find that the slopes are often smaller than expected, that they are almost always integers, that there is evidence of a connection between fractional slopes and slopes which are "bigger than usual", and that there are some hints of a connection to the theory of theta-cycles.
TL;DR: In this article, the homotopy groups of the Toda-Smith 4-cells spectrum were determined using the Bockstein spectral sequence (BSS) and the Adams-Novikov spectral sequence.
Abstract: At each prime number p, the homotopy groups p�…L2S 0 † of the v y1 2 BP- localized sphere spectrum play an crucial role to understand the category of v y1 2 BP- local spectra. For p > 3, they are determined by using the Adams-Novikov spectral sequence (ANSS), which collapses in this case. At the prime 3, p�…L2V…1†† is also determined by using the ANSS, in which Eyˆ E10 in this case. Here V…1† denotes the Toda-Smith 4-cells spectrum. In this paper, we determine the homotopy groups p�…L2V…0†† of the mod 3 Moore spectrum from p�…L2V…1†† by the Bockstein spectral sequence (BSS). Actually, we ®rst compute the E2-term of the ANSS by the BSS and then study the Adams-Novikov dierentials, and obtain EyE10 as well.
TL;DR: A review of the main developments since Lehmer's paper on integral sequences associated to polynomials can be found in this paper, where the distribution of primes in these sequences predicted by standard heuristic arguments.
Abstract: In a paper of 1933, D.H. Lehmer continued Pierce's study of integral sequences associated to polynomials, generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences -- or in closely related sequences -- would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness.
We review briefly some of the main developments since Lehmer's paper, and report on further computational work on these sequences. In particular, we use Mossinghoff's collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments.
The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.
Patent•
29 Mar 2000TL;DR: An interleaver in which a frame of data to be interleaved is stored in at least a portion of an array having R rows and C columns is defined in this paper.
Abstract: An interleaver in which a frame of data to be interleaved is stored in at least a portion of an array having R rows and C columns, the portion having Nr(1) rows and N?c?(1) columns that satisfy the inequality N?r??(1) x N?c(1-1) < L < N?r??(1) x N?c(1) where N?c?(1) is a prime number and N?c?(1-1) is the highest prime number less than N?c?(1). The elements of each row are permuted according to a predetermined mathematical relationship, and the rows are permuted according to predetermined mapping.
TL;DR: In this article, it was shown that there are infinitely many short intervals containing considerably more elements of the sum of two squares than expected, and infinitely many irregular intervals with considerably fewer than expected.
Abstract: LetS denote the set of integers representable as a sum of two squares. SinceS can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected thatS has many properties in common with the set of prime numbers. In this paper we exhibit "unexpected irregularities" in the distri- bution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements ofS than expected, and infinitely many intervals containing considerably fewer than expected.
1 Jan 2000
TL;DR: Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks as mentioned in this paper, such as the prime number theorem, which describes the asymptotic distribution of prime numbers.
Abstract: Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are:
$$\pi \left( x \right) \sim \frac{x} {{\log x}}as\quad x \to \infty ,$$
(1)
and
$${p_n} \sim \,n\,\log \,n\,as\,n\, \to \infty.$$
(2)
TL;DR: Under the assumption of the appropriate Ricmann hypothesis, it is shown in this article that the k-th power moment of certain normed error terms can be estimated effectively for x > x 0(q, e).
Abstract: Under the assumption of the appropriate Ricmann hypothesis it is shown that max
t≦x
min
l≠1
t
-1/2 (Ψ(x, q, 1) − Ψ(x, q, l)) > (
$$\frac{1}{2}$$
− e) log3
x and min
t≦x
max
l≠1
t
-1/2 (Ψ(x, q, 1) − Ψ(x, q, l)) < −(
$$\frac{1}{2}$$
− e) log3
x for x > x
0(q, e) The proof is quite elementary, and x
0 can be estimated effectively As a by-product a formula for the k-th power moment of certain normed error terms is obtained
1 Jan 2000
TL;DR: It is shown here that a simple evolutionary predator-prey model yields primeperiodic preys, and how a spatio-temporal extension of the model renders spiral waves being reminiscent of those observed in excitable systems, host-parasitoid systems and prebiotic evolution.
Abstract: In the present work, we shall allow for the merging of two seemingly unrelated subjects, one being periodical insects and the other the theory of prime numbers. The fact that some species of cicadas (genus Magicicada) appear every 7, 13 or 17 years and that these periods are prime numbers has been regarded as a coincidence [1, 2]. Without intending to argue in favour or against this statement, we will show here that a simple evolutionary predator-prey model yields primeperiodic preys. Moreover, this result will be used as a number-theoretical tool, namely to generate large prime numbers. Furthermore, we will demonstrate how a spatio-temporal extension of the model renders spiral waves being reminiscent of those observed in excitable systems, host-parasitoid systems and prebiotic evolution.
1 Jan 2000
TL;DR: Pósa as discussed by the authors showed that for all k > 1, there is an nk such that p1p2 · · · pn > p k n+1 for all n ≥ nk.
Abstract: and p1p2 · · · pn > p 3 n+1 for n ≥ 5. Stronger results of the same nature have been obtained by J. Sandór in [2]. For example p1p2 · · · pn > p 2 n+5 + p 2 [n/2] for n ≥ 24. Without the restrictions imposed by the use of elementary methods the precise determination of the margin from which the inequality holds, L. Pósa [3] proves the following result: For all k > 1 there is an nk such that p1p2 · · · pn > p k n+1 for all n ≥ nk.
TL;DR: In this paper, it was shown that G has no non-zero essential mod-p cohomology (and in fact that H * (G, IFp) is Cohen-Macaulay) if and only if IGI = 27 and exp(G) = 3.
Abstract: Let p be an odd prime number and let G be an extraspecial pgroup The purpose of the paper is to show that G has no non-zero essential mod-p cohomology (and in fact that H* (G, IFp) is Cohen-Macaulay) if and only if IGI = 27 and exp(G) = 3
1 Jan 2000
TL;DR: In this article, the authors give another proof of a result of R. Greenberg on the non-existence of non-trivial finite A-submodules of Selmer groups.
Abstract: In this note, we give another proof of a result of R. Greenberg on the non-existence of non-trivial finite A-submodules of Selmer groups. Let p be a prime number. Let K be a number field and E an elliptic curve defined over K. For any algebraic extension L/K and any place v of L, we denote by Lv the union of the completions at v of all finite extensions of K contained in L. We further denote by L (resp. Lv) a fixed algebraic closure of L (resp. Lv), and fix an immersion L -* Lv. Then the p'-Selmer group of E over L is defined as Selpoo (EIL) = Ker (H1 (Gal(L/L), Epoo) fJ H1 (Gal(Lv/Lv), E(Lv))),
TL;DR: In this paper, the Ultrafilter Principle was shown to be equivalent to the Boolean prime ideal theorem (or Ultrafilter principle) in arbitrary algebras with at least one binary operation, and it was shown that various separation lemmas and prime ideal theorems are special instances of one general theorem.
Abstract: We introduce ideals, radicals and prime ideals in arbitrary algebras with at least one binary operation, and we show that various separation lemmas and prime ideal theorems are special instances of one general theorem which, in turn, is equivalent to the Boolean Prime Ideal Theorem (or Ultrafilter Principle).
2 Jul 2000
TL;DR: This work describes a distributed implementation for computing the number of partitions with minimal space requirements and the resulting values are compared to those following from previously stated conjectures about the asymptotic behaviour of g.
Abstract: Computing the number of Goldbach partitions
$$g(n) = \#\{(p,q) | n = p + q, p \leq ~q\}$$
of all even numbers n up to a given limit can be done by a very simple, but space-demanding sequential procedure. This work describes a distributed implementation for computing the number of partitions with minimal space requirements. The program was distributed to numerous workstations, leading to the calculation of g(n) for all even n up to 5 × 108. The resulting values are compared to those following from previously stated conjectures about the asymptotic behaviour of g.
Patent•
14 Feb 2000TL;DR: In this article, a condition which can be intuitively hit upon, such as the bit length of a prime number or an extension degree, can be automatically generated, and a finite field operation can be performed using the expression data.
Abstract: If a condition which can be intuitively hit upon, such as the bit length of a prime number or an extension degree is designated, the expression data of a finite field corresponding to the condition can be automatically generated, and a finite field operation can be performed using the expression data.
1 Jan 2000
TL;DR: The Jacobian Conjecture has been extensively studied and various partial results have been obtained as mentioned in this paper, but this chapter will not be a collection of all these results and instead it will describe several new ways to attack the conjecture.
Abstract: As the reader certainly has noticed by now, the Jacobian Conjecture has been studied extensively and various partial results have been obtained This chapter will not be a collection of all these results Instead it will describe several new ways to attack the conjecture