Sieve Of Eratosthenes Research Articles

C++ for mathematicians: an introduction for students and professionals

List of Programs List of Figures Preface I. PROCEDURES The Basics What is C++? Hello C++ Numbers The Integer Types The Real Number Types The bool and char Types Checking the Size and Capacity of the Different Types Standard Operations Comparisons and Boolean Operations Complex Numbers Naming Variables Greatest Common Divisor The Problem A First Approach Euclid's Method Looping With for, while, and do An Exhaustive Approach to the GCD Problem Extended GCD, Call by Reference, and Overloading Random Numbers Pseudo Random Number Generation Uniform Random Values More on Pseudo Random Number Generation A Monte Carlo Program for the GCD Problem Normal Random Values Arrays Euler's Totient Array Fundamentals A Procedure to Factor Integers A Procedure to Calculate Euler's Totient The Sieve of Eratosthenes A Faster Totient Computing pn for Large n The Answer II. OBJECTS Points in the Plane Data and Methods Declaring the Point Class Data Hiding Constructors Assignment and Conversion Methods Procedures using Arguments of Type Point Operators Pythagorean Triples Generating Pythagorean Triples Designing a Primitive Pythagorean Triple Class Implementation of the PTriple Class Finding and Sorting the Triples Containers Sets Set Iterators Multisets Adjustable Arrays Via the Vector Class Ordered Pairs Maps Lists, Stacks, and Assorted Queues Modular Arithmetic Designing the Mod Type The Code The Default Modulus: Static Class Variables and Methods Constructors and Get/Set Methods Comparison Operators Arithmetic Operators Writing Mod Objects to Output Streams A Main to Demonstrate the Mod Class The Projective Plane Introduction to the Projective Plane, RP2 Designing the Classes PPoint and PLine Inheritance Protected Class Members Class and File Organization for PPoint and PLine The Parent Class PObject The Classes PPoint and PLine Discovering and Repairing a Bug Pappus Revisited Permutations Ulam's Problem Designing the Permutation Class Finding Monotone Subsequences Exercises Polynomials Procedure Templates Class Templates The Polynomial Class Template The GCD Problem Revisited Working in Binary III. TOPICS Using Other Packages Arbitrary Precision Arithmetic: the GMP Package Linear Algebra Other Packages Strings, Input/Output, and Visualization Character Arrays The String Class Command Line Arguments Reading and Writing Data in Files String Streams Formatting A Class to Parse Files Visualization Odds and Ends The switch Statement Labels and the goto Statement Exception Handling Friends Other Ways to Create Types Pointers IV. APPENDICES A. Your C++ Computing Environment Programming with a Command Window and a Text Editor Programming with an Integrated Development Environment General Advice on Debugging B. Documentation with Doxygen Doxygen Comments Using Doxygen C. C++ Reference Variables and Types Operations Control Statements Procedures Classes Standard Functions D. Answers Index *Each Chapter Contains Exercises Solutions can be found in Appendix D

Crystal structure of tveitite-(Y): Fractionation of rare-earth elements between positions and the variety of defects

The crystal structure of the mineral tveitite-(Y) (Y0.883Na0.106) (Ca0.841LREE0.159)(Ca0.716Na0.204HREE0.080)(Ca0.092Na0.074)F6.952 from amazonite pegmatites of Rov-Gora Mountain (Keivy, Kola Peninsula) is determined using X-ray diffraction (Stoe IPDS diffractometer, λMoKα, graphite monochromator, 2θmax = 63.5°, R = 0.051 for 1542 reflections). The main crystal data are as follows: a = 17.020(2) A, c = 9.679(2) A, V = 2428.2(4) A3, space group \(R\bar 3\), Z = 18, and ρcalcd = 4.00 g/cm3. The idealized structural formula of the mineral is represented as (Y,Na)6(Ca,LREE)6(Ca,Na,HREE)6(Ca,Na)F42 (Z = 3). The defect structure of the mineral manifests itself in a mixed occupation of all four independent cation positions and in a randomly disordered distribution of fluorine atoms over the majority of anion positions. It is shown that the crystal structure of tveitite-(Y) fulfills the function of an “Eratosthenes sieve” for yttrium cations and two groups of lanthanide cations, so that these cations are distributed over three different positions.

Lazy functional programming in Java

In this paper, we show how lazy functional programming techniques can be used within the Java programming language. We provide Java implementations of classic examples of lazy lists, such as the Sieve of Eratosthenes, the Eight Queens Problem, and natural-language parsing. We discuss how well these implementations succeed, compared to their original counterparts. We also point out the potential synergy between adding lazy techniques to Java, and adding generic types. The examples we provide would be suitable for teaching functional programming concepts in the context of a Java-based syllabus.

On twin primes associated with the Hawkins random sieve

We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59–81]. We here extend this to k ⩾ 2 and generalize it to all l-tuples of Hawkins primes.

Comments on the Misuse of Terminology in Savant Research: It is not the Sieve of Eratosthenes

Comments on the Misuse of Terminology in Savant Research: It is not the Sieve of Eratosthenes

Flowpaths: Compiling stack-based IR to hardware

The performance of software executed on a microprocessor is adversely affected by the basic fetch–execute cycle. A further performance penalty results from the load–execute–store paradigm associated with the use of local variables in most high-level languages. Implementing the software algorithm directly in hardware such as on an FPGA can alleviate these performance penalties. Such implementations are normally developed in a hardware description language such as VHDL or Verilog. More recently, several methods for using C as a hardware description language and for compiling C programs to hardware have been researched. Several software-programming languages compile to an intermediate representation (IR) that is stack based such as Java to Java bytecodes. Forth is a programming language that minimizes the use of local variables by exchanging the load–execute–store paradigm for stack manipulation instructions. This paper introduces a new systems architecture for FPGAs, called flowpaths, which can implement Java bytecodes or software programs written in Forth directly in an FPGA without the need for a microprocessor core. In the flowpath implementation of Forth programs all stack manipulation instructions are represented as simple wire connections that take zero time to execute. In the flowpath implementation of Java bytecodes the normal load–execute–store paradigm is represented as a single sequential operation and stack-manipulation operations become combinational thus executing faster. This paper compares the use of flowpaths in an FPGA generated from Java bytecodes and a high-level Forth program for the Sieve of Eratosthenes with C, Java, and Forth executed on microprocessors and microprocessor cores on FPGAs. The results show that flowpaths perform within a factor of two of a minimal hand-crafted direct hardware implementation of the Sieve and orders of magnitude better than compiling the program to a microprocessor.

Readers Write: March 2005

During summer 2004, a class of rising seventh graders in Alexandria, Virginia, constructed a “Sieve of Eratosthenes” for primes from 2 through 97. This Johns Hopkins Center for Talented Youth class then examined natural numbers from 3 to 100 using The Math Teacher's Book of Lists (Prentice Hall, 1995) to see if they had pairs of “symmetric primes” associated with them, which are obtained by adding or subtracting the same number.

The Lesser-Known Δ-Function in Number Theory

We actually use this definition for any nonnegative real number x and extend it to the entire real line by setting A(x, N) = A({x}, N). Here {x} = x [x] signifies the fractional part of x. As a function of x, A(x, N) is clearly periodic of period 1, with A(0, N) = 0. Large values of |A(x, N) I correspond to abnormal numbers of integers coprime to N in the interval [1, xN]. In particular, if N is the product of all primes up to some parameter y, such possible fluctuations of I A(x, N)I, as x varies, have a bearing on anticipated sieve bounds. These are modem versions of the classical sieve of Eratosthenes that produces primes, up to some fixed bound, from the list of positive integers

FUNCTIONAL PEARL Calculating the Sieve of Eratosthenes

The Sieve of Eratosthenes is an efficient algorithm for computing the successive primes. Rendered informally, it is as follows:1. Write down the successive “plurals”: 2, 3, 4, …2. Repeat:(a) Take the first number that is not circled or crossed out.(b) Circle it.(c) Cross out its proper multiples.3. What is left (i.e. the circled numbers) are the successive prime numbers.

A VHDL--Forth Core for FPGAs

The Forth programming language is typically implemented to run on some particular microprocessor. Several Forth engines have been designed that execute Forth instructions directly, typically in a single clock cycle. With the advent of high density FPGAs it has become feasible to implement a high-performance Forth core in an FPGA. This paper describes the design of a Forth core using VHDL that has been implemented on a Xilinx Spartan II FPGA. Examples are presented of high-level Forth programs that are compiled to VHDL code that implements a ROM embedded in the FPGA. The use of a Forth core in an FPGA allows for rapid prototyping of digital systems. Experiments show that an identical Forth program for the Sieve of Eratosthenes executes nearly 30 times faster on the FPGA Forth core than on a 68HC12 microcontroller at the same clock speed. This same program executes over six times faster on the FPGA Forth core than an equivalent compiled C program run on the same 68HC12. The Forth core is available as an EDIF file at www.tigs.com/fc16, which can be included in a VHDL project and uses approximately 30% of a Spartan II FPGA.

In Code: A Mathematical Journey

From the Publisher: In January 1999, Sarah Flannery, sports-loving teenager from Blarney in County Cork, Ireland, was awarded Ireland's Young Scientist of the Year for her extraordinary research and discoveries in Internet cryptography. The following day, her story began appearing in Irish papers and soon after was splashed across the front page of the London Times, complete with photo of Sarah and caption calling her brilliant. Just sixteen, she was mathematician with an international reputation. In Code is heartwarming story that will have readers cheering Sarah on. Originally published in England and cowritten with her mathematician father, David Flannery, In Code is a wonderfully moving story about the thrill of the mathematical chase (Nature) and a paean to intellectual adventure (Times Educational Supplement). A memoir in mathematics, it is all about how girl next door, nurtured by her family, moved from the simple math puzzles that were the staple of dinnertime conversation to prime numbers, the Sieve of Eratosthenes, Fermat's Little Theorem, googols--and finally into her breathtaking algorithm. Parallel with each step is modest girl's own self-discovery--her values, her burning curiosity, the joy of persistence, and, above all, her love for her family.

A Brief History of Factoring and Primality Testing B. C. (Before Computers)

Factoring and primality testing have become increasingly important in today's information based society, since they both have produced techniques used in the secure transmission of data. However, often lost in the modern-day shuffle of information are the contributions of the pioneers whose ideas ushered in the computer age and, as we shall see, some of whose ideas are still used today as the underpinnings of powerful algorithms for factoring and primality testing. We offer this brief history to help readers know more about these contributions and appreciate their significance. Virtually everyone who has graduated from high school knows the definition of a number, namely a p E N = {1, 2, 3, 4, . . .} such that p > 1 and if p = Em where X, m E , then either f = 1 or m = 1. (If E N and > 1 is not prime, then is called composite.) Although we cannot be certain, the concept of primality probably arose with the ancient Greeks over two and one-half millennia ago. The first recorded definition of numbers was given by Euclid around 300 BCE in his Elements. However, there is some indirect evidence that the concept of primality might have been known far earlier, for instance, to Pythagoras and his followers. The Greeks of antiquity used the term arithmetic to mean what today we would call number theory, namely the study of the properties of the natural numbers and the relationships between them. The Greeks reserved the word logistics for the study of ordinary computations using the standard operations of addition/subtraction and multiplication/division, which we now call arithmetic. The Pythagoreans introduced the term mathematics, which to them meant the study of arithmetic, astronomy, geometry, and music. This curriculum became known as the quadrivium in the Middle Ages. Although we have enjoyed the notion of a for millennia, only very recently have we developed eJficient tests for primality. This seemingly trivial task is in fact much more difficult than it appears. A primality test is an algorithm (a methodology following a set of rules to achieve a goal), the steps of which verify that given some integer n, we may conclude n is a number. A primality proof is a successful application of a primality test. Such tests are typically called true primality tests to distinguish them from probabilistic primality tests (which can only conclude that n is prime up to a specified likelihood). We will not discuss such algorithms here (see [9] for these). A concept used frequently in primality testing is the notion of a sieve. A is a process to find numbers with particular characteristics (for instance primes) by searching among all integers up to a prescribed bound, and eliminating invalid candidates until only the desired numbers remain. Eratosthenes (ca. 284-204 scE) proposed the first sieve for finding primes. The following example illustrates the Sieve of Eratosthenes.

A course in computational number theory

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.

The Combinatorics behind Number-Theoretic Sieves

Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated refinements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or Möbius inversion) are applicable. Unfortunately, these two developments have proceeded largely independently of each other even though they are closely related. This paper begins the process of bridging the gap between them by showing that much of the theory behind the number-theoretic refinements carries over readily to many combinatorial settings. The hope is that this will result in new approaches to and more powerful tools for sieve problems in combinatorics such as the computation of chromatic polynomials, the enumeration of permutations with restricted position, and the enumeration of regions in hyperplane arrangements.

Tasking solutions to the Sieve of Eratosthenes

Shapiro and later Bokhari described a multiprocessing view of the Sieve of Eratosthenes, to test load balancing on shared-memory multiprocessors. Shapiro proposed the following parallel version of the sieve:Eratosthenes starts off with a large number of slaves and instructs the first one to write down the integers 2, 3, 4, … on the ground in a field. A second slave is dispatched with instructions to cross out all multiples of 2. Eratosthenes now moves down the "list" of numbers and dispatches a new slave whenever an uncrossed number is encountered. Thus, the third slave is sent off with instructions to cross out the multiples of 3, the fourth crosses out all multiples of 5, and so on. The uncrossed numbers that remain at the end of this process are the primes.It turns out that Ada (especially Ada95) supports multiprocessing well in the Sieve of Eratosthenes. One may make use of static and dynamic tasks, protected types, communication, and synchronization in a relatively small program, making the sieve an excellent choice for instruction. In a graduate course we offer at William & Mary, Theory of Programming Languages, we give students a multiprocessing assignment. At this stage, students have rarely done parallel programming, except for low-level threads and locks in operating systems courses. So a parallel programming assignment in a high level language adds an extra tool to their arsenals and gives them an idea of an important part of computing today.

An algorithm for constructing a variety of arbitrary finite dimension

The dimension of a variety V of algebras is the greatest length of a basis (i.e., of an independent generating set) for an SC-theory SC(V), consisting of strong Mal'tsev conditions satisfied in V. The variety V is assumed infinite-dimensional if the lengths of the bases in SC(V) are not bounded. A simple algorithm is found for constructing a variety of any finite dimension r≥1. Using the sieve of Eratosthenes, r distinct primes p1, p2,…, pr are written and their product n=p1p2…pr computed. The variety Gn of algebras (A, f) with one n-ary operation satisfying the identity f(x1, x2,…,xn)=f(x2,…,xn, x1) has, then, dimension r.

Recent Improvements to the Sieve of Eratosthenes

The development and growth of electronic transmissions of information as well as the need for computertesting algorithms are partly responsible for the resurgence of activity in the area of prime numbers that we have witnessed in the last thirty years. In particular, the search for efficient algorithms to generate tables of prime numbers has produced excellent new results (Bengelloum 1986; Mairson 1977; Pritchard 1981). It has also produced noticeable improvements of such traditional algorithms as the sieve of Eratosthenes. This article presents recently developed generalizations to the sieve of Eratosthenes, showing the principles underlying these improvements, which increase its efficiency without changing too much of its simplicity.

FUNCTIONAL PEARL Lazy wheel sieves and spirals of primes

The popular method of enumerating the primes is the Sieve of Eratosthenes. It can be programmed very neatly in a lazy functional language, but runs rather slowly. A little-known alternative method is the Wheel Sieve, originally formulated as a fast imperative algorithm for obtaining all primes up to a given limit, assuming destructive access to a bit-array. This article describes functional variants of the wheel sieve that enumerate all primes as a lazy list.

Distribution of integer partitions with large number of summands

The factor c−1 √ n log n is nothing but the asymptotic value of the mean number of summands in a random partition of n, each partition of n being equally likely; cf. [17]. Note that the number of partitions of n with m 1’s is given by p(n −m) − p(n −m − 1) for 0 ≤ m ≤ n; and, consequently, p(n−m) is nothing but the number of partitions of n which have ≥ m 1’s. Thus the factor one plays an important role in the counting function p(n,m) when m is larger than the mean value. Our aim in this paper is to show that such a phenomenon holds for more general partitions. The proof of the theorem is rather simple and relies on a method that we previously employed in [10] (for Dirichlet series) for an analytic proof of a result of Nicolas [15]. From the formula (2), it is obvious that our method is based on explicitly isolating the contribution of the factor 1 to the counting function p(n,m). This method is quite general and can be applied to other partition problems; it can be regarded as analytic version of the sieve of Eratosthenes; cf. [14, Ch. IV]. A general theorem under a scheme due to Meinardus [13] will be derived in § 4, which applies, in particular, to partitions into powers. Finally, we further refine the analysis to establish the limiting distribution (with convergence rate) for the number of summands in partitions into parts ≥ k, 1 ≤ k = o ( n1/4(log n)−1 ) , thus extending and improving the results by Auluck et al. [2]. Notation. We shall use the notation [zn]f(z) to represent the coefficient of zn in the Taylor expansion of f(z). The notation [umzn]g(u, z) is then defined as [um] ([zn]g(u, z)). All limits (including O, , o and ∼), whenever unspecified, will be taken to be n→ +∞. The constant c always denotes √ 2/3π.

Exploring prime numbers within a spreadsheet

This paper presents the use of a spreadsheet in generating prime numbers through the Sieve of Eratosthenes. The approach is applied to visualizing frequency and irregularity in the occurrence of prime numbers, distribution of the primes gaps, and the behaviour of the prime counting function π(x).It is argued that these and other aspects of the theory of numbers are more easily represented on a spreadsheet than in a programming language. The spreadsheet used is Excel 4.0 for Macintosh computers.