Showing papers on "Prime number published in 1985"
Book•
1 Jan 1985
TL;DR: In this paper, Mapes' algorithm is used to construct a compact prime table, which is then used to test compositeness of numbers of the form N = h * 2n +-k.
Abstract: 1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning Computer Programs.- A Sieve Program.- Compact Prime Tables.- Hexadecimal Compact Prime Tables.- Difference Between Consecutive Primes.- The Number of Primes Below x.- Meissel's Formula.- Evaluation of Pk(x, a).- Lehmer's Formula.- Computations.- A Computation Using Meissel's Formula.- A Computation Using Lehmer's Formula.- A Computer Program Using Lehmer's Formula.- Mapes' Method.- Deduction of Formulas.- A Worked Example.- Mapes' Algorithm.- Programming Mapes' Algorithm.- Recent Developments.- Results.- Computational Complexity.- Comparison Between the Methods Discussed.- 2. The Primes Viewed at Large.- No Polynomial Can Produce Only Primes.- Formulas Yielding All Primes.- The Distribution of Primes Viewed at Large. Euclid's Theorem.- The Formulas of Gauss and Legendre for ?(x). The Prime Number Theorem.- The Chebyshev Function ?(x).- The Riemann Zeta-function.- The Zeros of the Zeta-function.- Conversion From f(x) Back to ?(x).- The Riemann Prime Number Formula.- The Sign of li x ? ?(x).- The Influence of the Complex Zeros of ?(s) on ?(x).- The Remainder Term in the Prime Number Theorem.- Effective Inequalities for ?(x), pn, and ?(x).- The Number of Primes in Arithmetic Progressions.- 3. Subtleties in the Distribution of Primes.- The Distribution of Primes in Short Intervals.- Twins and Some Other Constellations of Primes.- Admissible Constellations of Primes.- The Hardy-Littlewood Constants.- The Prime k-Tuples Conjecture.- Theoretical Evidence in Favour of the Prime k-Tuples Conjecture.- Numerical Evidence in Favour of the Prime k-Tuples Conjecture.- The Second Hardy-Littlewood Conjecture.- The Midpoint Sieve.- Modification of the Midpoint Sieve.- Construction of Superdense Admissible Constellations.- Some Dense Clusters of Primes.- The Distribution of Primes Between the Two Series 4n + 1 and 4n + 3.- Graph of the Function ?4,3(x) ? ?4,1(x).- The Negative Regions.- The Negative Blocks.- Large Gaps Between Consecutive Primes.- The Cramer Conjecture.- 4. The Recognition of Primes.- Tests of Primality and of Compositeness.- Factorization Methods as Tests of Compositeness.- Fermat's Theorem as Compositeness Test.- Fermat's Theorem as Primality Test.- Pseudoprimes and Probable Primes.- A Computer Program for Fermat's Test.- The Labor Involved in a Fermat Test.- Carmichael Numbers.- Euler Pseudoprimes.- Strong Pseudoprimes and a Primality Test.- A Computer Program for Strong Pseudoprime Tests.- Counts of Pseudoprimes and Carmichael Numbers.- Rigorous Primality Proofs.- Lehmer's Converse of Fermat's Theorem.- Formal Proof of Theorem 4.3.- Ad Hoc Search for a Primitive Root.- The Use of Several Bases.- Fermat Numbers and Pepin's Theorem.- Cofactors of Fermat Numbers.- Generalized Fermat Numbers.- A Relaxed Converse of Fermat's Theorem.- Proth's Theorem.- Tests of Compositeness for Numbers of the form N = h * 2n +- k.- An Alternative Approach.- Certificates of Primality.- Primality Tests of Lucasian Type.- Lucas Sequences.- The Fibonacci Numbers.- Large Subscripts.- An Alternative Deduction.- Divisibility Properties of the Numbers Un.- Primality Proofs by Aid of Lucas Sequences.- Lucas Tests for Mersenne Numbers.- A Relaxation of Theorem 4.8.- Pocklington's Theorem.- Lehmer-Pocklington's Theorem.- Pocklington-Type Theorems for Lucas Sequences.- Primality Tests for Integers of the form N = h * 2n ? 1, when 3?h.- Primality Tests for N = h * 2n ? 1, when 3?h.- The Combined N ? 1 and N + 1 Test.- Lucas Pseudoprimes.- Modern Primality Proofs.- The Jacobi Sum Primality Test.- Three Lemmas.- Lenstra's Theorem.- The Sets P and Q.- Running Time for the APRCL Test.- Elliptic Curve Primality Proving, ECPP.- The Goldwasser-Kilian Test.- Atkin's Test.- 5. Classical Methods of Factorization.- When Do We Attempt Factorization?.- Trial Division.- A Computer Implementation of Trial Division.- Euclid's Algorithm as an Aid to Factorization.- Fermat's Factoring Method.- Legendre's Congruence.- Euler's Factoring Method.- Gauss' Factoring Method.- Legendre's Factoring Method.- The Number of Prime Factors of Large Numbers.- How Does a Typical Factorization Look?.- The Erd?s-Kac Theorem.- The Distribution of Prime Factors of Various Sizes.- Dickman's Version of Theorem 5.4.- A More Detailed Theory.- The Size of the kth Largest Prime Factor of N.- Smooth Integers.- Searching for Factors of Certain Forms.- Legendre's Theorem for the Factors of N = an +- bn.- Adaptation to Search for Factors of the Form p = 2kn + 1.- Adaptation of Trial Division.- Adaptation of Fermat's Factoring Method.- Adaptation of Euclid's Algorithm as an Aid to Factorization.- 6. Modem Factorization Methods.- Choice of Method.- Pollard's (p ? 1)-Method.- Phase 2 of the (p ? 1)-Method.- The (p + 1)-Method.- Pollard's rho Method.- A Computer Program for Pollard's rho Method.- An Algebraic Description of Pollard's rho Method.- Brent's Modification of Pollard's rho Method.- The Pollard-Brent Method for p = 2kn + 1.- Shanks' Factoring Method SQUFOF.- A Computer Program for SQUFOF.- Comparison Between Pollard's rho Method and SQUFOF.- Morrison and Brillhart's Continued Fraction Method CFRAC.- The Factor Base.- An Example of a Factorization with CFRAC.- Further Details of CFRAC.- The Early Abort Strategy.- Results Achieved with CFRAC.- Running Time Analysis of CFRAC.- The Quadratic Sieve, QS.- Smallest Solutions to Q(x) ? 0 mod p.- Special Factors.- Results Achieved with QS.- The Multiple Polynomial Quadratic Sieve, MPQS.- Results Achieved with MPQS.- Running Time Analysis of QS and MPQS.- The Schnorr-Lenstra Method.- Two Categories of Factorization Methods.- Lenstra's Elliptic Curve Method, ECM.- Phase 2 of ECM.- The Choice of A, B, and P1.- Running Times of ECM.- Recent Results Achieved with ECM.- The Number Field Sieve, NFS.- Factoring Both in Z and in Z(z).- A Numerical Example.- The General Number Field Sieve, GNFS.- Running Times of NFS and GNFS.- Results Achieved with NFS. Factorization of F9.- Strategies in Factoring.- How Fast Can a Factorization Algorithm Be?.- 7. Prime Numbers and Cryptography.- Practical Secrecy.- Keys in Cryptography.- Arithmetical Formulation.- RSA Cryptosystems.- How to Find the Recovery Exponent.- A Worked Example.- Selecting Keys.- Finding Suitable Primes.- The Fixed Points of an RSA System.- How Safe is an RSA Cryptosystem?.- Superior Factorization.- Appendix 1. Basic Concepts in Higher Algebra.- Modules.- Euclid's Algorithm.- The Labor Involved in Euclid's Algorithm.- A Definition Taken from the Theory of Algorithms.- A Computer Program for Euclid's Algorithm.- Reducing the Labor.- Binary Form of Euclid's Algorithm.- Groups.- Lagrange's Theorem. Cosets.- Abstract Groups. Isomorphic Groups.- The Direct Product of Two Given Groups.- Cyclic Groups.- Rings.- Zero Divisors.- Fields.- Mappings. Isomorphisms and Homomorphisms.- Group Characters.- The Conjugate or Inverse Character.- Homomorphisms and Group Characters.- Appendix 2. Basic Concepts in Higher Arithmetic.- Divisors. Common Divisors.- The Fundamental Theorem of Arithmetic.- Congruences.- Linear Congruences.- Linear Congruences and Euclid's Algorithm.- Systems of Linear Congruences.- Carmichael's Function.- Carmichael's Theorem.- Appendix 3. Quadratic Residues.- Legendre's Symbol.- Arithmetic Rules for Residues and Non-Residues.- The Law of Quadratic Reciprocity.- Jacobi's Symbol.- Appendix 4. The Arithmetic of Quadratic Fields.- Appendix 5. Higher Algebraic Number Fields.- Algebraic Numbers.- Appendix 6. Algebraic Factors.- Factorization of Polynomials.- The Cyclotomic Polynomials.- Aurifeuillian Factorizations.- Factorization Formulas.- The Algebraic Structure of Aurifeuillian Numbers.- Appendix 7. Elliptic Curves.- Cubics.- Rational Points on Rational Cubics.- Homogeneous Coordinates.- Elliptic Curves.- Rational Points on Elliptic Curves.- Appendix 8. Continued Fractions.- What Is a Continued Fraction?.- Regular Continued Fractions. Expansions.- Evaluating a Continued Fraction.- Continued Fractions as Approximations.- Euclid's Algorithm and Continued Fractions.- Linear Diophantine Equations and Continued Fractions.- A Computer Program.- Continued Fraction Expansions of Square Roots.- Proof of Periodicity.- The Maximal Period-Length.- Short Periods.- Continued Fractions and Quadratic Residues.- Appendix 9. Multiple-Precision Arithmetic.- Various Objectives for a Multiple-Precision Package.- How to Store Multi-Precise Integers.- Addition and Subtraction of Multi-Precise Integers.- Reduction in Length of Multi-Precise Integers.- Multiplication of Multi-Precise Integers.- Division of Multi-Precise Integers.- Input and Output of Multi-Precise Integers.- A Complete Package for Multiple-Precision Arithmetic.- A Computer Program for Pollard's rho Method.- Appendix 10. Fast Multiplication of Large Integers.- The Ordinary Multiplication Algorithm.- Double Length Multiplication.- Recursive Use of Double Length Multiplication Formula.- A Recursive Procedure for Squaring Large Integers.- Fractal Structure of Recursive Squaring.- Large Mersenne Primes.- Appendix 11. The Stieltjes Integral.- Functions With Jump Discontinuities.- The Riemann Integral.- Definition of the Stieltjes Integral.- Rules of Integration for Stieltjes Integrals.- Integration by Parts of Stieltjes Integrals.- The Mean Value Theorem.- Applications.- Tables. For Contents.- List of Textbooks.
478 citations
1 Dec 1985
TL;DR: It is shown that the problem of finding strong, random,Large primes is only 19% harder than finding random, large primes.
Abstract: A simple method is given for finding strong, random, large primes of a given number of bits, for use in conjunction with the RSA Public Key Cryptosystem. A strong prime p is a prime satisfying: * p = 1 mod r * p = s-1 mod s * r = 1 mod t, where r,s and t are all large, random primes of a given number of bits. It is shown that the problem of finding strong, random, large primes is only 19% harder than finding random, large primes.
80 citations
TL;DR: An algorithm for computing discrete logarithms over GF (p^{2}) , where p is a prime, in subexponential time is described, which uses quadratic fields as the appropriate algebraic structure.
Abstract: An algorithm for computing discrete logarithms over GF (p^{2}) , where p is a prime, in subexponential time is described. The algorithm is similar to the Merkle-Adleman algorithm for computing logarithms over GF (p) , but it uses quadratic fields as the appropriate algebraic structure. It also makes use of the idea of a virtual spanning set due to Hellman and Reyneri for computing discrete logarithms over GF (p^{m}) , for m growing and p fixed.
68 citations
1 Jan 1985
TL;DR: An implementation of the Cohen-Lenstra version (see ibid., vol. 42, p.297-330, 1984) of the Adleman-Pomerance-Rumely primality test was presented as mentioned in this paper.
Abstract: An implementation of the Cohen-Lenstra version (see ibid., vol. 42, p.297-330, 1984) of the Adleman-Pomerance-Rumely primality test (see L.M. Adleman, C. Pomerance and R.S. Rumely, Ann. of Math., vol.117, p.173-206, 1983) is presented. Primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes
65 citations
37 citations
Book•
1 Jan 1985
TL;DR: In a recent article as discussed by the authors, the authors describe what a mathematician does and why: "What does a mathematician do and why? Prime Numbers, Diophantine equations, and great problems of geometry and space".
Abstract: What does a mathematician do and why? Prime Numbers.- A lively activity: To do mathematics Diophantine equations.- Great problems of geometry and space.
21 citations
1 Jan 1985
TL;DR: In this paper, it was shown that if d denotes the degree of the jacobian of the mapping (P,Q), then each of the coordinates P and Q has at most d + 2 zeros at infinity.
Abstract: In the paper there have been investigated polynomial mappings (P,Q): ℂ2 → ℂ2 in the aspect of the connection between the structure of the jacobian and the coordinates of the mapping. There have been obtained some informations on coefficients of an expansion of the Newton-Puiseux type of one of the coordinates with respect to the other one. Starting from these informations, a theorem is proved stating that if d denotes the degree of the jacobian of the mapping (P,Q), then each of the coordinates P and Q has at most d + 2 zeros at infinity. There have been obtained some equations connecting the homogeneous components of the polynomials P and Q.
16 citations
15 citations
TL;DR: In this paper, the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup is discussed, where the relation is extended to complex multiplication.
Abstract: This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted by Φ(x) = 0.
14 citations
Journal Article•
8 citations
1 Jan 1985
Abstract: We show that over the algebraic closure of a finite field, every point of the jacobian of a curve annihilated by a power of a prime I is the I-primary component of a point in the image of the curve. Let X be a smooth, projective, geometrically connected curve of genus g > 0 defined over the algebraic closure F of the field Fq of q elements. Fixing a basepoint x( of X in Fq, let p: X -* J denote the embedding assigning to each point x of X the divisor class of the difference of x and x0. Let I be any prime number and let X: J(Fq) J(Fq), denote the projection of the torsion group J(Fq) onto its i-primary component. The object of this note is to prove the following THEOREM. The map X o q: X(Fq) J(F) is surjective. For the proof we need a lemma giving control over the distribution of primes of degree one in arithmetic progressions. Let K/k be an abelian unramified extension of global fields of positive characteristic. Let g be the genus of k and suppose that Fq is the field of constants of both K and k. For each prime v of k let Ft, E Gal(K/k) denote the corresponding arithmetic Frobenius. LEMMA. If there exists a0 E Gal(K/k) such that F,, / a0 for all primes v of k of degree one, then q < (2g[ K: k] + 3) 2. PROOF. By hypothesis (* )E = -E 4(co) ,(Ftj) C1 C1 4 where v runs through all the primes of k of degree one (i.e., of residue field coinciding with Fq), and 4 runs through all the nontrivial complex-valued characters of Gal(K/k). Now by the Riemann Hypothesis (see Appendix 5 of [W]) the left side of (*) is bounded below by q + 1 2g~/F; the right side is bounded above by ([K: k] 1)(2g 2)v/Vj. The desired conclusion follows immediately. Turning now to the proof of the theorem, suppose that some point d E J(Fq) fails to be in the image of X o T. Assume, as is permissible, that X is the base-change of a smooth projective curve X0 defined over Fq, x0 is Fq-rational, and d is an Fq-rational point of the jacobian J0 of X0. Fix a rational prime r distinct from 1. Let k denote the function field of X0, k an algebraic closure of k, and v0 the prime of degree one of k Received by the editors April 3, 1984. 1980 Mathemnatics Subject Classificcation. Primary 12A80; Secondary 14H99. 'A1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page
TL;DR: In this article, the authors present a general divisibility algorithm for prime numbers that can be used to test a nonnegative integer n is divisible by any of the primes 2, 3, 5, and 11.
Abstract: Many people are interested in divisibility rules, particularly since they can be coordinated with calculator activities For example, young students can use calculators to gather data, examine the data for patterns, and discover some divisibility rules on their own (See Bitter and Mikesell 1980; Reys et al 1979) Generally, calculators afford a quick way to determine if a number is divisible by a certain prime Divisibility rules are particularly useful when the number being tested for divisibility is too large for a calculator to handle exactly In this article I present a neat and general divisibility algorithm for prime numbers This procedure is significant because it can be used to test a number for divisibility by almost every prime Rules and procedures for testing whether a nonnegative integer n is divisible by any of the primes 2, 3, 5, and 11 are well known and easy to apply That is, n is divisible by -
TL;DR: In this paper, the authors consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over F p and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.
Abstract: Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p . It is easy to see and more or less known that for any natural number n , there exists an elliptic curve over p whose j -invariant is of degree n over F p and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over F p and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f .
TL;DR: In this paper, it was shown that all graphs on a non-prime number v of vertices are α,β destructible for some a which divides v. This is called stable.
Abstract: A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are integral factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in a β-set V'. Graphs which are not α, β destructible for any α.β are called stable. In this paper we prove that all graphs on a non-prime number v of vertices are α,β destructible for some a which divides v.
1 Jan 1985
TL;DR: It is concluded that the application of the prime number approach leads to significant increases in speed and some reduction in storage requirements.
Abstract: Computational sequencing of nucleic acid and amino acid sequences is placing increasing demands on computer resources. The use of prime numbers is explored as a convenient means of improving program speed and reducing storage requirements. It is concluded that the application of the prime number approach leads to significant increases in speed and some reduction in storage requirements.
6 Nov 1985
TL;DR: The prime factor Fourier transform system arestructed based on a new designed parallel processing array processor called Vector Engine and compared with some well known FFT algorithms, the performance analysis is discussed.
Abstract: A new method for implementing prime factor discrete Fourier transforms on array processors is presented. 'The prime factor Fourier transform system are mnstructed based on a new designed parallel processing array processor called Vector Engine. Basic architecture for short length Prime number Fourier transform implementations are discussed. By applying the short length architecture, the implemelltation of long length prime factor Fourier tansforms are discussed and designed. Compared with some well known FFT algorithms, the performance analysis of this system is also discussed.
1 Jan 1985
TL;DR: In this paper, the distinction between pure mathematics and applied mathematics was discussed, and the aesthetic side of mathematics was also discussed, with a focus on why mathematicians do mathematics and why they like it.
Abstract: The conference started with why, for ten minutes. I do mathematics because I like it. We discussed briefly the distinction between pure mathematics and applied mathematics, which actually intermingle in such a way that it is impossible to define the boundary between one and the other precisely; and also the aesthetic side of mathematics. Then we did mathematics together. I started by defining prime numbers, and I recalled Euclid’s proof that there are infinitely many. Then I defined twin primes, (3,5), (5, 7), (11,13), (17,19), etc. which differ by 2. Is there an infinite number of those? No one knows, even though one conjectures that the answer is yes. I gave heuristic arguments describing the expected density of such primes. Why don’t you try to prove it? The question is one of the big unsolved problems of mathematics.
01 Mar 1985-Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering
TL;DR: Signal flow graphs for networks of electronic systems are often reduced to transfer functions by Mason's theorem and an interactive software package is presented to make this process fully automated.
Abstract: Signal flow graphs for networks of electronic systems are often reduced to transfer functions by Mason's theorem An interactive software package is presented to make this process fully automated The nodes are numbered Input to the program specifies the transmittance between node pairs, first noting the start‐node‐number followed by that of the destination‐node Transmittances are represented either as real numbers or as a string of characters With real transmittances the output of the program takes the form of a numerator and a denominator of the transfer function, each as a real number With character‐string transmittances the numerator and denominator of the output are available both as strings and as arrays of products of prime numbers, where each prime number has uniquely represented an input character string
1 Jan 1985
TL;DR: In this paper, Groupe d'étude en théorie analytique des nombres implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Abstract: © Groupe d’étude en théorie analytique des nombres (Secrétariat mathématique, Paris), 1984-1985, tous droits réservés. L’accès aux archives de la collection « Groupe d’étude en théorie analytique des nombres » 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.
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TL;DR: In this paper, the authors constructed rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime.
Abstract: In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let