DOCSLIB.ORG

(Mersenne Primes Search) 11191649 Jun Li June, 2012

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Institute of Information and Mathematical Sciences Prime Number Search Algorithms (Mersenne Primes Search) 11191649 Jun Li June, 2012 Contents CHAPTER 1 INTRODUCTION ......................................................................................................... 1 1.1 BACKGROUND .................................................................................................................................... 1 1.2 MERSENNE PRIME ............................................................................................................................. 1 1.3 STUDY HISTORY ................................................................................................................................ 2 1.3.1 Early history [4] ......................................................................................................................... 2 1.3.2 Modern History .......................................................................................................................... 3 1.3.3 Recent History ............................................................................................................................ 4 CHAPTER 2 METHODOLOGY ........................................................................................................ 6 2.1 DEFINITION AND THEOREMS ............................................................................................................. 6 2.2 DISTRIBUTION LAW ........................................................................................................................... 7 2.3 ALGORITHMS ..................................................................................................................................... 8 2.3.1 Trial Division ............................................................................................................................. 8 2.3.2 Sieve of Eratosthenes ................................................................................................................. 9 2.3.3 Pollard's (P-1) method ............................................................................................................. 12 2.3.4 Lucas-Lehmer Test ................................................................................................................... 12 2.4 GREAT INTERNET MERSENNE PRIME SEARCH .............................................................................. 13 2.4.1 A little bit about GIMPS ........................................................................................................... 13 2.4.2 How GIMPS Works .................................................................................................................. 14 2.4.3 GIMPS and Grid Computing ................................................................................................... 16 CHAPTER 3 EXPERIMENT AND RESULTS ................................................................................. 18 3.1 “LUCAS-LEHMER TEST WITH GMP” AND “GLUCAS”.................................................................... 18 3.1.1 Lucas-Lehmer test with GMP .................................................................................................. 18 3.1.2 Glucas ....................................................................................................................................... 18 3.1.3 “Lucas-Lehmer test with GMP” vs “Glucas” ......................................................................... 19 3.1.4 Experiment conclusion ............................................................................................................ 23 3.2 UNDISCOVERED BETWEEN MERSENNE PRIMES 41ST AND 42ND .................................................... 25 3.2.1 Basic method ............................................................................................................................ 25 3.2.2 Experiment ............................................................................................................................... 25 3.2.3 Conclusion ............................................................................................................................... 26 3.3 STUDY MERSENNE PRIMES ON GPU .............................................................................................. 27 3.3.1 Introduction to GPU ................................................................................................................ 27 3.3.2 GPU Computing ....................................................................................................................... 27 3.3.3 Experiments on GPU ............................................................................................................... 28 3.3.4 Conclusion ............................................................................................................................... 35 CHAPTER 4 DISCUSSION AND FUTURE WORK ....................................................................... 36 4.1 SUMMARY OF EXPERIMENTS ........................................................................................................... 36 4.2 MEANING OF STUDY MERSENNE PRIMES ....................................................................................... 36 4.3 FUTURE WORK ................................................................................................................................. 37 REFERENCES .................................................................................................................................. 39 APPENDIX ........................................................................................................................................ 41 A. PROGRAMS .................................................................................................................................... 41 B. TABLES .......................................................................................................................................... 43 C. FIGURES ........................................................................................................................................ 43 Chapter 1 Introduction Although math doesn't bring excitement breakthrough very often, sometimes we can still hear the news that mathematicians discover a new super-large prime number. Those numbers are usually with a form which called Mersenne primes. 1.1 Background On April 12th, 2009, the 47th known Mersenne prime number——242,643,801 − 1 was found by Odd Magnar Strindmo, a Norwegian IT professional, through Great Internet Mersenne Prime Search (GIMPS). It was confirmed by Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman on June 7, 2009. The 47th Mersenne prime number is a 12,837,064 digit number, the second largest Mersenne prime number, which will be more than 50 kilometers if we use ordinary font size to write it down. [1] 1.2 Mersenne Prime There are some concepts and their relationship we have to be clear before we start our project: Prime Number, Mersenne Number and Mersenne Prime. Prime Number: In mathematics, a prime number, according to wolframalpha, that “A positive integer that has exactly one positive integer divisor other than 1 (i.e., no factors other than 1 and itself.)”[2] For example, 2, 3, 5, 7, 11 are prime numbers, as only 1 and themselves can divide them. However 4 is composite, since it has the divisors 2 and 2 in addition to 1 and 4. Usually people just call prime numbers as primes. Mersenne Number: In mathematics, a Mersenne number, named after Marin Mersenne (a French theologian and mathematician, who began the study of these numbers in the early 17th century, but known for his work on acoustics), is a number of the form, 푝 푀푝 = 2 − 1, where p is an integer. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255 and so on. Mersenne Prime: In mathematics, a Mersenne prime is a Mersenne number that is prime. The form is just like below, 1 푝 푀푝 = 2 − 1, where Mp and p are primes. Marin Mersenne compiled a list of primes in the form of 2푝 − 1 (exponent p≤257), but it is wrongly included M67 and M257 which are not primes and missed M61, M89 and M107. [3] The Mersenne number is not necessarily a prime number, the following examples show very clear that Mersenne number can be a prime number or a non-prime number. 2 3 Prime number:푀2 = 2 − 1 = 3 푀3 = 2 − 1 = 7 4 Non-prime number: 푀4 = 2 − 1 = 15 1.3 Study History Mersenne prime was first proposed in order to solve perfect number. The history of searching Mersenne prime can be traced back to 350 BC. However, even till today, human beings have found only 47 Mersenne primes. Twelve of them are found before 1952 by mankind using pen and paper to calculate. The rest are found by computer. 1.3.1 Early history [4] Over 2300 years ago, Euclid, the ancient Greek mathematician, started to study a number of the form 2푝 − 1 (the first time people study about 2푝 − 1in history) when he discussed about perfect number in his book "Euclid's Elements". In the 17th century, the famous French mathematician, founder of the French Academy of Sciences, Marin Mersenne (1588-1648), the first people who studied 2푝 − 1 deeply and systematically. To commemorate him, the mathematical community named the number of this form 푝 푀푝 = 2 − 1 (Mp and p are primes) as Mersenne primes. As the core issue of number theory, Mersenne number can be described as the focus of research. It has been a thought that for all n, the number in the form of 2푛 − 1 is a prime number. This is obviously wrong. There are new
Recommended publications
  • Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster

    Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster

    THE OPEN BOOK SERIES 2 ANTS XIII Proceedings of the Thirteenth Algorithmic Number Theory Symposium Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster msp THE OPEN BOOK SERIES 2 (2019) Thirteenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2019.2.411 Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW-challenge pseudoprimes with two or three prime factors up to 280. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods. 1. Introduction Pomerance, Selfridge, and Wagstaff famously offered $620 for a composite n that satisfies (1) 2n 1 1 .mod n/ so n is a base-2 Fermat pseudoprime, Á (2) .5 n/ 1 so n is not a square modulo 5, and j D (3) Fn 1 0 .mod n/ so n is a Fibonacci pseudoprime, C Á or to prove that no such n exists. We call composites that satisfy these conditions PSW-challenge pseudo- primes. In[PSW80] they credit R. Baillie with the discovery that combining a Fermat test with a Lucas test (with a certain specific parameter choice) makes for an especially effective primality test[BW80].
  • SIMD Extensions

    SIMD Extensions

    SIMD Extensions PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sat, 12 May 2012 17:14:46 UTC Contents Articles SIMD 1 MMX (instruction set) 6 3DNow! 8 Streaming SIMD Extensions 12 SSE2 16 SSE3 18 SSSE3 20 SSE4 22 SSE5 26 Advanced Vector Extensions 28 CVT16 instruction set 31 XOP instruction set 31 References Article Sources and Contributors 33 Image Sources, Licenses and Contributors 34 Article Licenses License 35 SIMD 1 SIMD Single instruction Multiple instruction Single data SISD MISD Multiple data SIMD MIMD Single instruction, multiple data (SIMD), is a class of parallel computers in Flynn's taxonomy. It describes computers with multiple processing elements that perform the same operation on multiple data simultaneously. Thus, such machines exploit data level parallelism. History The first use of SIMD instructions was in vector supercomputers of the early 1970s such as the CDC Star-100 and the Texas Instruments ASC, which could operate on a vector of data with a single instruction. Vector processing was especially popularized by Cray in the 1970s and 1980s. Vector-processing architectures are now considered separate from SIMD machines, based on the fact that vector machines processed the vectors one word at a time through pipelined processors (though still based on a single instruction), whereas modern SIMD machines process all elements of the vector simultaneously.[1] The first era of modern SIMD machines was characterized by massively parallel processing-style supercomputers such as the Thinking Machines CM-1 and CM-2. These machines had many limited-functionality processors that would work in parallel.
  • Fast Generation of RSA Keys Using Smooth Integers

    Fast Generation of RSA Keys Using Smooth Integers

    1 Fast Generation of RSA Keys using Smooth Integers Vassil Dimitrov, Luigi Vigneri and Vidal Attias Abstract—Primality generation is the cornerstone of several essential cryptographic systems. The problem has been a subject of deep investigations, but there is still a substantial room for improvements. Typically, the algorithms used have two parts – trial divisions aimed at eliminating numbers with small prime factors and primality tests based on an easy-to-compute statement that is valid for primes and invalid for composites. In this paper, we will showcase a technique that will eliminate the first phase of the primality testing algorithms. The computational simulations show a reduction of the primality generation time by about 30% in the case of 1024-bit RSA key pairs. This can be particularly beneficial in the case of decentralized environments for shared RSA keys as the initial trial division part of the key generation algorithms can be avoided at no cost. This also significantly reduces the communication complexity. Another essential contribution of the paper is the introduction of a new one-way function that is computationally simpler than the existing ones used in public-key cryptography. This function can be used to create new random number generators, and it also could be potentially used for designing entirely new public-key encryption systems. Index Terms—Multiple-base Representations, Public-Key Cryptography, Primality Testing, Computational Number Theory, RSA ✦ 1 INTRODUCTION 1.1 Fast generation of prime numbers DDITIVE number theory is a fascinating area of The generation of prime numbers is a cornerstone of A mathematics. In it one can find problems with cryptographic systems such as the RSA cryptosystem.
  • Grade 7/8 Math Circles the Scale of Numbers Introduction

    Grade 7/8 Math Circles the Scale of Numbers Introduction

    Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero.
  • The Exponential Function

    The Exponential Function

    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 5-2006 The Exponential Function Shawn A. Mousel University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Mousel, Shawn A., "The Exponential Function" (2006). MAT Exam Expository Papers. 26. https://digitalcommons.unl.edu/mathmidexppap/26 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. The Exponential Function Expository Paper Shawn A. Mousel In partial fulfillment of the requirements for the Masters of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor May 2006 Mousel – MAT Expository Paper - 1 One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax , where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x 2. One way to recognize the difference between the two functions is by the name of the function.
  • Simple Statements, Large Numbers

    Simple Statements, Large Numbers

    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 Simple Statements, Large Numbers Shana Streeks University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Streeks, Shana, "Simple Statements, Large Numbers" (2007). MAT Exam Expository Papers. 41. https://digitalcommons.unl.edu/mathmidexppap/41 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Master of Arts in Teaching (MAT) Masters Exam Shana Streeks In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Gordon Woodward, Advisor July 2007 Simple Statements, Large Numbers Shana Streeks July 2007 Page 1 Streeks Simple Statements, Large Numbers Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion.
  • The Quadratic Sieve Factoring Algorithm

    The Quadratic Sieve Factoring Algorithm

    The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 1 Introduction Mathematicians have been attempting to find better and faster ways to fac- tor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factoriza- tion. This trial division was not improved upon until Fermat applied the factorization of the difference of two squares: a2 b2 = (a b)(a + b). In his method, we begin with the number to be factored:− n. We− find the smallest square larger than n, and test to see if the difference is square. If so, then we can apply the trick of factoring the difference of two squares to find the factors of n. If the difference is not a perfect square, then we find the next largest square, and repeat the process. While Fermat's method is much faster than trial division, when it comes to the real world of factoring, for example factoring an RSA modulus several hundred digits long, the purely iterative method of Fermat is too slow. Sev- eral other methods have been presented, such as the Elliptic Curve Method discovered by H. Lenstra in 1987 and a pair of probabilistic methods by Pollard in the mid 70's, the p 1 method and the ρ method. The fastest algorithms, however, utilize the− same trick as Fermat, examples of which are the Continued Fraction Method, the Quadratic Sieve (and it variants), and the Number Field Sieve (and its variants). The exception to this is the El- liptic Curve Method, which runs almost as fast as the Quadratic Sieve.
  • The Notion Of" Unimaginable Numbers" in Computational Number Theory

    The Notion Of" Unimaginable Numbers" in Computational Number Theory

    Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers.
  • Hyperoperations and Nopt Structures

    Hyperoperations and Nopt Structures

    Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles.
  • Standards Chapter 10: Exponents and Scientific Notation Key Terms

    Standards Chapter 10: Exponents and Scientific Notation Key Terms

    Chapter 10: Exponents and Scientific Notation Standards Common Core: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical Essential Questions Students will… expressions. How can you use exponents to Write expressions using write numbers? integer exponents. 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate How can you use inductive Evaluate expressions very large or very small quantities, and to express reasoning to observe patterns involving integer exponents. how many times as much one is than the other. and write general rules involving properties of Multiply powers with the 8.EE.4: Perform operations with numbers expressed exponents? same base. in scientific notation, including problems where both decimal and scientific notation are used. Use How can you divide two Find a power of a power. scientific notation and choose units of appropriate powers that have the same size for measurements of very large or very small base? Find a power of a product. quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been How can you evaluate a Divide powers with the same generated by technology. nonzero number with an base. exponent of zero? How can you evaluate a nonzero Simplify expressions number with a negative integer involving the quotient of Key Terms exponent? powers. A power is a product of repeated factors. How can you read numbers Evaluate expressions that are written in scientific involving numbers with zero The base of a power is the common factor.
  • Cryptology and Computational Number Theory (Boulder, Colorado, August 1989) 41 R

    Cryptology and Computational Number Theory (Boulder, Colorado, August 1989) 41 R

    http://dx.doi.org/10.1090/psapm/042 Other Titles in This Series 50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R.
  • Newsletter 115, 2017

    Newsletter 115, 2017

    Cumann Oid´ıMatamaitice na hEireann´ N e w s l e t t e r Irish Mathematics Teachers' Association 2017 Number 115 Maths Development Team Important update From September 1st 2017, the Maths Development Team(MDT) will form part of the Professional Development Service for Teachers. We will continue to provide professional development to post-primary teachers of Mathematics through the following supports: ◦ The effective use of Lesson Study: All new participants are asked to sign up at (https://goo.gl/forms/GJRS7GLuHdehVZqz1) for one of our three-day Induction Programmes, which will be held in Dublin, Carrick-on-Shannon and Limerick on September 27th , 28th and 29th , 2017. Closing date for receipt of applications: Monday 11th September 2017. ◦ A series of six Lesson Study meetings to be held from October 2017 to February 2018. These meetings will be conducted in over 50 locations nationally. Contact Administrator Gráinne Haughney: email [email protected] Phone: 01- 8576422 for venues. ◦ Ongoing participation in Lesson Study for all mathematics teachers. ◦ On-line support and resources available at www.projectmaths.ie ◦ Half-day school visits to support the on-going professional development of teachers. The primary focus will be on teaching and learning using reflective approaches with particular emphasis on: i. Algebra through the Lens of Functions, ii. Geometry through the Lens of Enquiry, iii. School-identified needs. Please note that substitution will not be provided for teachers availing of these school visits. If you wish to apply for a member of the team to visit your Mathematics Department during the 2017-2018 school year, please contact [email protected] or [email protected] for an application form.